$sl{}\left(2\right)$ -subalgebras of sp(14), type $C_7$

sp(14), type

C_7

Structure constants and notation.
Root subalgebras / root subsystems.
sl(2)-subalgebras.

Page generated by the calculator project.

Number of sl(2) subalgebras: 63.

Let

h

be in the Cartan subalgebra. Let

\left({\alpha}_{1}, \dots, {\alpha}_{n}\right)

be simple roots with respect to

h

. Then the

h

-characteristic, as defined by E. Dynkin, is the

n

-tuple

\left({\alpha}_{1}{}\left(h\right), \dots, {\alpha}_{n}{}\left(h\right)\right)

The actual realization of h. The coordinates of

h

are given with respect to the fixed original simple basis. Note that the

h

-characteristic is computed using a possibly different simple basis, more precisely, with respect to any h-positive simple basis.

A regular semisimple subalgebra might contain an

sl{}\left(2\right)

such that it has no centralizer in the regular semisimple subalgebra, but the regular semisimple subalgebra might fail to be minimal containing. This happens when another minimal containing regular semisimple subalgebra of equal rank nests as a root subalgebra in the containing subalgebra. See Dynkin, Semisimple Lie subalgebras of semisimple Lie algebras, remark before Theorem 10.4.

The

sl{}\left(2\right)

submodules of the ambient Lie algebra are parametrized by their highest weight with respect to the Cartan element

h

sl{}\left(2\right)

. In turn, the highest weight is a positive integer multiple of the fundamental highest weight

\psi

{V}_{k \psi}

k+1

-dimensional.

Type + realization link	h-Characteristic	Realization of h	sl(2)-module decomposition of the ambient Lie algebra $\psi$ = the fundamental $sl{}\left(2\right)$ -weight.	Centralizer dimension	Type of semisimple part of centralizer, if known	The square of the length of the weight dual to h.	Dynkin index	Minimal containing regular semisimple SAs	Containing regular semisimple SAs in which the sl(2) has no centralizer
$A^{455}_1$	(2, 2, 2, 2, 2, 2, 2)	(26, 48, 66, 80, 90, 96, 49)	$V_{26\psi}+V_{22\psi}+V_{18\psi}+V_{14\psi}+V_{10\psi}+V_{6\psi}+V_{2\psi}$	0	$0$	910	455	$C^{1}_7$	$C^{1}_7$
$A^{287}_1$	(2, 2, 2, 2, 2, 0, 2)	(22, 40, 54, 64, 70, 72, 37)	$V_{22\psi}+V_{18\psi}+V_{14\psi}+V_{12\psi}+2V_{10\psi}+V_{6\psi}+2V_{2\psi}$	0	$0$	574	287	$C^{1}_6+A^{1}_1$	$C^{1}_6+A^{1}_1$ , $C^{1}_7$
$A^{286}_1$	(2, 2, 2, 2, 2, 1, 0)	(22, 40, 54, 64, 70, 72, 36)	$V_{22\psi}+V_{18\psi}+V_{14\psi}+2V_{11\psi}+V_{10\psi}+V_{6\psi}+V_{2\psi}+3V_{0}$	3	$A^{1}_1$	572	286	$C^{1}_6$	$C^{1}_6$
$A^{175}_1$	(2, 2, 2, 0, 2, 0, 2)	(18, 32, 42, 48, 54, 56, 29)	$V_{18\psi}+V_{14\psi}+V_{12\psi}+2V_{10\psi}+V_{8\psi}+3V_{6\psi}+2V_{2\psi}$	0	$0$	350	175	$C^{1}_5+B^{1}_2$	$C^{1}_5+B^{1}_2$ , $C^{1}_7$
$A^{167}_1$	(2, 2, 2, 2, 0, 0, 2)	(18, 32, 42, 48, 50, 52, 27)	$V_{18\psi}+V_{14\psi}+3V_{10\psi}+2V_{8\psi}+V_{6\psi}+4V_{2\psi}+V_{0}$	1	$0$	334	167	$C^{1}_5+2A^{1}_1$ , $C^{1}_5+A^{2}_1$	$C^{1}_5+2A^{1}_1$ , $C^{1}_5+A^{2}_1$ , $C^{1}_6+A^{1}_1$
$A^{166}_1$	(2, 2, 2, 2, 0, 1, 0)	(18, 32, 42, 48, 50, 52, 26)	$V_{18\psi}+V_{14\psi}+2V_{10\psi}+2V_{9\psi}+V_{8\psi}+V_{6\psi}+2V_{2\psi}+2V_{\psi}+3V_{0}$	3	$A^{1}_1$	332	166	$C^{1}_5+A^{1}_1$	$C^{1}_5+A^{1}_1$ , $C^{1}_6$
$A^{165}_1$	(2, 2, 2, 2, 1, 0, 0)	(18, 32, 42, 48, 50, 50, 25)	$V_{18\psi}+V_{14\psi}+V_{10\psi}+4V_{9\psi}+V_{6\psi}+V_{2\psi}+10V_{0}$	10	$B^{1}_2$	330	165	$C^{1}_5$	$C^{1}_5$
$A^{119}_1$	(2, 0, 2, 0, 2, 0, 2)	(14, 24, 34, 40, 46, 48, 25)	$V_{14\psi}+V_{12\psi}+3V_{10\psi}+V_{8\psi}+3V_{6\psi}+V_{4\psi}+3V_{2\psi}$	0	$0$	238	119	$C^{1}_4+C^{1}_3$	$C^{1}_4+C^{1}_3$ , $C^{1}_7$
$A^{112}_1$	(0, 2, 0, 2, 0, 2, 0)	(12, 24, 32, 40, 44, 48, 24)	$3V_{12\psi}+V_{10\psi}+3V_{8\psi}+V_{6\psi}+3V_{4\psi}+V_{2\psi}+3V_{0}$	3	$A^{7}_1$	224	112	$A^{2}_6$	$A^{2}_6$
$A^{95}_1$	(2, 2, 0, 2, 0, 0, 2)	(14, 24, 30, 36, 38, 40, 21)	$V_{14\psi}+2V_{10\psi}+2V_{8\psi}+4V_{6\psi}+2V_{4\psi}+4V_{2\psi}$	0	$0$	190	95	$C^{1}_4+B^{1}_2+A^{1}_1$	$C^{1}_4+B^{1}_2+A^{1}_1$ , $C^{1}_7$ , $C^{1}_6+A^{1}_1$ , $C^{1}_5+B^{1}_2$ , $C^{1}_4+C^{1}_3$
$A^{94}_1$	(2, 2, 0, 2, 0, 1, 0)	(14, 24, 30, 36, 38, 40, 20)	$V_{14\psi}+2V_{10\psi}+V_{8\psi}+2V_{7\psi}+3V_{6\psi}+V_{4\psi}+2V_{3\psi}+2V_{2\psi}+3V_{0}$	3	$A^{1}_1$	188	94	$C^{1}_4+B^{1}_2$	$C^{1}_4+B^{1}_2$ , $C^{1}_6$
$A^{92}_1$	(2, 2, 1, 0, 1, 1, 0)	(14, 24, 30, 34, 38, 40, 20)	$V_{14\psi}+V_{10\psi}+2V_{9\psi}+2V_{7\psi}+V_{6\psi}+2V_{5\psi}+3V_{4\psi}+2V_{2\psi}+3V_{0}$	3	$A^{3}_1$	184	92	$C^{1}_4+A^{2}_2$	$C^{1}_4+A^{2}_2$
$A^{87}_1$	(2, 2, 2, 0, 0, 0, 2)	(14, 24, 30, 32, 34, 36, 19)	$V_{14\psi}+V_{10\psi}+3V_{8\psi}+4V_{6\psi}+7V_{2\psi}+3V_{0}$	3	$A^{8}_1$	174	87	$C^{1}_4+3A^{1}_1$ , $C^{1}_4+A^{2}_1+A^{1}_1$	$C^{1}_4+3A^{1}_1$ , $C^{1}_4+A^{2}_1+A^{1}_1$ , $C^{1}_5+2A^{1}_1$ , $C^{1}_5+A^{2}_1$
$A^{86}_1$	(2, 2, 2, 0, 0, 1, 0)	(14, 24, 30, 32, 34, 36, 18)	$V_{14\psi}+V_{10\psi}+2V_{8\psi}+2V_{7\psi}+3V_{6\psi}+4V_{2\psi}+4V_{\psi}+4V_{0}$	4	$A^{1}_1$	172	86	$C^{1}_4+2A^{1}_1$ , $C^{1}_4+A^{2}_1$	$C^{1}_4+2A^{1}_1$ , $C^{1}_4+A^{2}_1$ , $C^{1}_5+A^{1}_1$
$A^{85}_1$	(2, 2, 2, 0, 1, 0, 0)	(14, 24, 30, 32, 34, 34, 17)	$V_{14\psi}+V_{10\psi}+V_{8\psi}+4V_{7\psi}+2V_{6\psi}+2V_{2\psi}+4V_{\psi}+10V_{0}$	10	$B^{1}_2$	170	85	$C^{1}_4+A^{1}_1$	$C^{1}_4+A^{1}_1$ , $C^{1}_5$
$A^{84}_1$	(2, 2, 2, 1, 0, 0, 0)	(14, 24, 30, 32, 32, 32, 16)	$V_{14\psi}+V_{10\psi}+6V_{7\psi}+V_{6\psi}+V_{2\psi}+21V_{0}$	21	$C^{1}_3$	168	84	$C^{1}_4$	$C^{1}_4$
$A^{71}_1$	(0, 2, 0, 2, 0, 0, 2)	(10, 20, 26, 32, 34, 36, 19)	$3V_{10\psi}+V_{8\psi}+5V_{6\psi}+3V_{4\psi}+4V_{2\psi}+V_{0}$	1	$0$	142	71	$2C^{1}_3+A^{1}_1$ , $A^{2}_5+A^{1}_1$	$2C^{1}_3+A^{1}_1$ , $A^{2}_5+A^{1}_1$ , $C^{1}_4+C^{1}_3$
$A^{70}_1$	(0, 2, 0, 2, 0, 1, 0)	(10, 20, 26, 32, 34, 36, 18)	$3V_{10\psi}+V_{8\psi}+3V_{6\psi}+4V_{5\psi}+V_{4\psi}+3V_{2\psi}+4V_{0}$	4	$A^{1}_1$	140	70	$2C^{1}_3$ , $A^{2}_5$	$2C^{1}_3$ , $A^{2}_5$
$A^{55}_1$	(2, 0, 0, 2, 0, 0, 2)	(10, 16, 22, 28, 30, 32, 17)	$V_{10\psi}+2V_{8\psi}+6V_{6\psi}+3V_{4\psi}+6V_{2\psi}+V_{0}$	1	$0$	110	55	$C^{1}_3+2B^{1}_2$ , $A^{2}_3+C^{1}_3$	$C^{1}_3+2B^{1}_2$ , $A^{2}_3+C^{1}_3$ , $C^{1}_5+B^{1}_2$
$A^{50}_1$	(0, 1, 1, 0, 1, 1, 0)	(8, 16, 22, 26, 30, 32, 16)	$3V_{8\psi}+2V_{7\psi}+2V_{6\psi}+2V_{5\psi}+3V_{4\psi}+2V_{3\psi}+2V_{2\psi}+2V_{\psi}+3V_{0}$	3	$A^{5}_1$	100	50	$A^{2}_4+B^{1}_2$	$A^{2}_4+B^{1}_2$
$A^{47}_1$	(2, 0, 2, 0, 0, 0, 2)	(10, 16, 22, 24, 26, 28, 15)	$V_{10\psi}+V_{8\psi}+5V_{6\psi}+5V_{4\psi}+8V_{2\psi}+V_{0}$	1	$0$	94	47	$C^{1}_3+B^{1}_2+2A^{1}_1$ , $C^{1}_3+B^{1}_2+A^{2}_1$	$C^{1}_3+B^{1}_2+2A^{1}_1$ , $C^{1}_3+B^{1}_2+A^{2}_1$ , $C^{1}_6+A^{1}_1$ , $C^{1}_5+2A^{1}_1$ , $C^{1}_4+C^{1}_3$ , $C^{1}_4+B^{1}_2+A^{1}_1$ , $2C^{1}_3+A^{1}_1$ , $C^{1}_5+A^{2}_1$
$A^{46}_1$	(2, 0, 2, 0, 0, 1, 0)	(10, 16, 22, 24, 26, 28, 14)	$V_{10\psi}+V_{8\psi}+4V_{6\psi}+2V_{5\psi}+3V_{4\psi}+2V_{3\psi}+5V_{2\psi}+2V_{\psi}+3V_{0}$	3	$A^{1}_1$	92	46	$C^{1}_3+B^{1}_2+A^{1}_1$	$C^{1}_3+B^{1}_2+A^{1}_1$ , $C^{1}_6$ , $C^{1}_5+A^{1}_1$ , $C^{1}_4+B^{1}_2$ , $2C^{1}_3$
$A^{45}_1$	(2, 0, 2, 0, 1, 0, 0)	(10, 16, 22, 24, 26, 26, 13)	$V_{10\psi}+V_{8\psi}+3V_{6\psi}+4V_{5\psi}+V_{4\psi}+4V_{3\psi}+3V_{2\psi}+10V_{0}$	10	$B^{1}_2$	90	45	$C^{1}_3+B^{1}_2$	$C^{1}_3+B^{1}_2$ , $C^{1}_5$
$A^{44}_1$	(2, 1, 0, 1, 0, 1, 0)	(10, 16, 20, 24, 26, 28, 14)	$V_{10\psi}+2V_{7\psi}+2V_{6\psi}+2V_{5\psi}+4V_{4\psi}+4V_{3\psi}+3V_{2\psi}+2V_{\psi}+3V_{0}$	3	$A^{3}_1$	88	44	$C^{1}_3+A^{2}_2+A^{1}_1$	$C^{1}_3+A^{2}_2+A^{1}_1$ , $C^{1}_4+A^{2}_2$
$A^{43}_1$	(2, 1, 0, 1, 1, 0, 0)	(10, 16, 20, 24, 26, 26, 13)	$V_{10\psi}+2V_{7\psi}+V_{6\psi}+4V_{5\psi}+3V_{4\psi}+2V_{3\psi}+6V_{2\psi}+6V_{0}$	6	$A^{3}_1+A^{1}_1$	86	43	$C^{1}_3+A^{2}_2$	$C^{1}_3+A^{2}_2$
$A^{42}_1$	(0, 2, 0, 1, 0, 1, 0)	(8, 16, 20, 24, 26, 28, 14)	$3V_{8\psi}+V_{6\psi}+4V_{5\psi}+3V_{4\psi}+4V_{3\psi}+4V_{2\psi}+4V_{0}$	4	$A^{5}_1$	84	42	$A^{2}_4+2A^{1}_1$ , $A^{2}_4+A^{2}_1$	$A^{2}_4+2A^{1}_1$ , $A^{2}_4+A^{2}_1$
$A^{41}_1$	(0, 2, 0, 1, 1, 0, 0)	(8, 16, 20, 24, 26, 26, 13)	$3V_{8\psi}+V_{6\psi}+2V_{5\psi}+7V_{4\psi}+2V_{3\psi}+2V_{2\psi}+2V_{\psi}+6V_{0}$	6	$A^{5}_1+A^{1}_1$	82	41	$A^{2}_4+A^{1}_1$	$A^{2}_4+A^{1}_1$
$A^{40}_1$	(0, 2, 0, 2, 0, 0, 0)	(8, 16, 20, 24, 24, 24, 12)	$3V_{8\psi}+V_{6\psi}+11V_{4\psi}+V_{2\psi}+13V_{0}$	13	$B^{1}_2+A^{5}_1$	80	40	$A^{2}_4$	$A^{2}_4$
$A^{39}_1$	(2, 2, 0, 0, 0, 0, 2)	(10, 16, 18, 20, 22, 24, 13)	$V_{10\psi}+5V_{6\psi}+4V_{4\psi}+11V_{2\psi}+6V_{0}$	6	$2A^{4}_1$	78	39	$C^{1}_3+4A^{1}_1$ , $C^{1}_3+A^{2}_1+2A^{1}_1$ , $C^{1}_3+2A^{2}_1$	$C^{1}_3+4A^{1}_1$ , $C^{1}_3+A^{2}_1+2A^{1}_1$ , $C^{1}_3+2A^{2}_1$ , $C^{1}_4+3A^{1}_1$ , $C^{1}_4+A^{2}_1+A^{1}_1$
$A^{38}_1$	(2, 2, 0, 0, 0, 1, 0)	(10, 16, 18, 20, 22, 24, 12)	$V_{10\psi}+4V_{6\psi}+2V_{5\psi}+3V_{4\psi}+7V_{2\psi}+6V_{\psi}+6V_{0}$	6	$A^{8}_1+A^{1}_1$	76	38	$C^{1}_3+3A^{1}_1$ , $C^{1}_3+A^{2}_1+A^{1}_1$	$C^{1}_3+3A^{1}_1$ , $C^{1}_3+A^{2}_1+A^{1}_1$ , $C^{1}_4+2A^{1}_1$ , $C^{1}_4+A^{2}_1$
$A^{37}_1$	(2, 2, 0, 0, 1, 0, 0)	(10, 16, 18, 20, 22, 22, 11)	$V_{10\psi}+3V_{6\psi}+4V_{5\psi}+2V_{4\psi}+4V_{2\psi}+8V_{\psi}+11V_{0}$	11	$B^{1}_2$	74	37	$C^{1}_3+2A^{1}_1$ , $C^{1}_3+A^{2}_1$	$C^{1}_3+2A^{1}_1$ , $C^{1}_3+A^{2}_1$ , $C^{1}_4+A^{1}_1$
$A^{36}_1$	(2, 2, 0, 1, 0, 0, 0)	(10, 16, 18, 20, 20, 20, 10)	$V_{10\psi}+2V_{6\psi}+6V_{5\psi}+V_{4\psi}+2V_{2\psi}+6V_{\psi}+21V_{0}$	21	$C^{1}_3$	72	36	$C^{1}_3+A^{1}_1$	$C^{1}_3+A^{1}_1$ , $C^{1}_4$
$A^{35}_1$	(2, 2, 1, 0, 0, 0, 0)	(10, 16, 18, 18, 18, 18, 9)	$V_{10\psi}+V_{6\psi}+8V_{5\psi}+V_{2\psi}+36V_{0}$	36	$C^{1}_4$	70	35	$C^{1}_3$	$C^{1}_3$
$A^{31}_1$	(0, 0, 2, 0, 0, 0, 2)	(6, 12, 18, 20, 22, 24, 13)	$6V_{6\psi}+6V_{4\psi}+10V_{2\psi}+3V_{0}$	3	$A^{16}_1$	62	31	$3B^{1}_2+A^{1}_1$ , $A^{2}_3+B^{1}_2+A^{1}_1$	$3B^{1}_2+A^{1}_1$ , $A^{2}_3+B^{1}_2+A^{1}_1$ , $C^{1}_3+2B^{1}_2$ , $A^{2}_3+C^{1}_3$
$A^{30}_1$	(0, 0, 2, 0, 0, 1, 0)	(6, 12, 18, 20, 22, 24, 12)	$6V_{6\psi}+3V_{4\psi}+6V_{3\psi}+6V_{2\psi}+6V_{0}$	6	$A^{16}_1+A^{1}_1$	60	30	$3B^{1}_2$ , $A^{2}_3+B^{1}_2$	$3B^{1}_2$ , $A^{2}_3+B^{1}_2$
$A^{28}_1$	(0, 1, 0, 1, 0, 1, 0)	(6, 12, 16, 20, 22, 24, 12)	$3V_{6\psi}+4V_{5\psi}+4V_{4\psi}+4V_{3\psi}+4V_{2\psi}+4V_{\psi}+4V_{0}$	4	$A^{3}_1$	56	28	$A^{2}_2+2B^{1}_2$ , $A^{2}_3+A^{2}_2$	$A^{2}_2+2B^{1}_2$ , $A^{2}_3+A^{2}_2$
$A^{23}_1$	(0, 2, 0, 0, 0, 0, 2)	(6, 12, 14, 16, 18, 20, 11)	$3V_{6\psi}+7V_{4\psi}+15V_{2\psi}+4V_{0}$	4	$A^{8}_1$	46	23	$2B^{1}_2+3A^{1}_1$ , $A^{2}_3+3A^{1}_1$ , $2B^{1}_2+A^{2}_1+A^{1}_1$ , $A^{2}_3+A^{2}_1+A^{1}_1$	$2B^{1}_2+3A^{1}_1$ , $A^{2}_3+3A^{1}_1$ , $2B^{1}_2+A^{2}_1+A^{1}_1$ , $A^{2}_3+A^{2}_1+A^{1}_1$ , $2C^{1}_3+A^{1}_1$ , $C^{1}_3+B^{1}_2+2A^{1}_1$ , $C^{1}_3+B^{1}_2+A^{2}_1$
$A^{22}_1$	(0, 2, 0, 0, 0, 1, 0)	(6, 12, 14, 16, 18, 20, 10)	$3V_{6\psi}+5V_{4\psi}+4V_{3\psi}+10V_{2\psi}+4V_{\psi}+5V_{0}$	5	$A^{1}_1$	44	22	$2B^{1}_2+2A^{1}_1$ , $A^{2}_3+2A^{1}_1$ , $2B^{1}_2+A^{2}_1$ , $A^{2}_3+A^{2}_1$	$2B^{1}_2+2A^{1}_1$ , $A^{2}_3+2A^{1}_1$ , $2B^{1}_2+A^{2}_1$ , $A^{2}_3+A^{2}_1$ , $2C^{1}_3$ , $C^{1}_3+B^{1}_2+A^{1}_1$
$A^{21}_1$	(0, 2, 0, 0, 1, 0, 0)	(6, 12, 14, 16, 18, 18, 9)	$3V_{6\psi}+3V_{4\psi}+8V_{3\psi}+6V_{2\psi}+4V_{\psi}+11V_{0}$	11	$B^{1}_2$	42	21	$2B^{1}_2+A^{1}_1$ , $A^{2}_3+A^{1}_1$	$2B^{1}_2+A^{1}_1$ , $A^{2}_3+A^{1}_1$ , $C^{1}_3+B^{1}_2$
$A^{20}_1$	(1, 0, 1, 0, 0, 1, 0)	(6, 10, 14, 16, 18, 20, 10)	$V_{6\psi}+2V_{5\psi}+5V_{4\psi}+6V_{3\psi}+7V_{2\psi}+6V_{\psi}+4V_{0}$	4	$A^{3}_1$	40	20	$A^{2}_2+B^{1}_2+2A^{1}_1$ , $A^{2}_2+B^{1}_2+A^{2}_1$	$A^{2}_2+B^{1}_2+2A^{1}_1$ , $A^{2}_2+B^{1}_2+A^{2}_1$ , $C^{1}_3+A^{2}_2+A^{1}_1$
$A^{20}_1$	(0, 2, 0, 1, 0, 0, 0)	(6, 12, 14, 16, 16, 16, 8)	$3V_{6\psi}+V_{4\psi}+12V_{3\psi}+3V_{2\psi}+22V_{0}$	22	$C^{1}_3$	40	20	$2B^{1}_2$ , $A^{2}_3$	$2B^{1}_2$ , $A^{2}_3$
$A^{19}_1$	(1, 0, 1, 0, 1, 0, 0)	(6, 10, 14, 16, 18, 18, 9)	$V_{6\psi}+2V_{5\psi}+4V_{4\psi}+6V_{3\psi}+8V_{2\psi}+6V_{\psi}+6V_{0}$	6	$A^{3}_1+A^{1}_1$	38	19	$A^{2}_2+B^{1}_2+A^{1}_1$	$A^{2}_2+B^{1}_2+A^{1}_1$ , $C^{1}_3+A^{2}_2$
$A^{18}_1$	(1, 0, 1, 1, 0, 0, 0)	(6, 10, 14, 16, 16, 16, 8)	$V_{6\psi}+2V_{5\psi}+3V_{4\psi}+6V_{3\psi}+10V_{2\psi}+2V_{\psi}+13V_{0}$	13	$B^{1}_2+A^{3}_1$	36	18	$A^{2}_2+B^{1}_2$	$A^{2}_2+B^{1}_2$
$A^{17}_1$	(0, 0, 0, 1, 1, 0, 0)	(4, 8, 12, 16, 18, 18, 9)	$10V_{4\psi}+4V_{3\psi}+7V_{2\psi}+4V_{\psi}+10V_{0}$	10	$B^{3}_2$	34	17	$2A^{2}_2+A^{1}_1$	$2A^{2}_2+A^{1}_1$
$A^{16}_1$	(0, 0, 0, 2, 0, 0, 0)	(4, 8, 12, 16, 16, 16, 8)	$10V_{4\psi}+14V_{2\psi}+13V_{0}$	13	$B^{3}_2+A^{1}_1$	32	16	$2A^{2}_2$	$2A^{2}_2$
$A^{15}_1$	(2, 0, 0, 0, 0, 0, 2)	(6, 8, 10, 12, 14, 16, 9)	$V_{6\psi}+5V_{4\psi}+21V_{2\psi}+10V_{0}$	10	$B^{4}_2$	30	15	$B^{1}_2+2A^{2}_1+A^{1}_1$ , $B^{1}_2+5A^{1}_1$ , $B^{1}_2+A^{2}_1+3A^{1}_1$	$B^{1}_2+2A^{2}_1+A^{1}_1$ , $C^{1}_3+4A^{1}_1$ , $B^{1}_2+5A^{1}_1$ , $C^{1}_3+A^{2}_1+2A^{1}_1$ , $B^{1}_2+A^{2}_1+3A^{1}_1$ , $C^{1}_3+2A^{2}_1$
$A^{14}_1$	(2, 0, 0, 0, 0, 1, 0)	(6, 8, 10, 12, 14, 16, 8)	$V_{6\psi}+4V_{4\psi}+2V_{3\psi}+15V_{2\psi}+8V_{\psi}+9V_{0}$	9	$2A^{4}_1+A^{1}_1$	28	14	$B^{1}_2+4A^{1}_1$ , $B^{1}_2+A^{2}_1+2A^{1}_1$ , $B^{1}_2+2A^{2}_1$	$B^{1}_2+4A^{1}_1$ , $B^{1}_2+A^{2}_1+2A^{1}_1$ , $B^{1}_2+2A^{2}_1$ , $C^{1}_3+3A^{1}_1$ , $C^{1}_3+A^{2}_1+A^{1}_1$
$A^{13}_1$	(2, 0, 0, 0, 1, 0, 0)	(6, 8, 10, 12, 14, 14, 7)	$V_{6\psi}+3V_{4\psi}+4V_{3\psi}+10V_{2\psi}+12V_{\psi}+13V_{0}$	13	$B^{1}_2+A^{8}_1$	26	13	$B^{1}_2+3A^{1}_1$ , $B^{1}_2+A^{2}_1+A^{1}_1$	$B^{1}_2+3A^{1}_1$ , $B^{1}_2+A^{2}_1+A^{1}_1$ , $C^{1}_3+2A^{1}_1$ , $C^{1}_3+A^{2}_1$
$A^{12}_1$	(2, 0, 0, 1, 0, 0, 0)	(6, 8, 10, 12, 12, 12, 6)	$V_{6\psi}+2V_{4\psi}+6V_{3\psi}+6V_{2\psi}+12V_{\psi}+22V_{0}$	22	$C^{1}_3$	24	12	$B^{1}_2+2A^{1}_1$ , $B^{1}_2+A^{2}_1$	$B^{1}_2+2A^{1}_1$ , $B^{1}_2+A^{2}_1$ , $C^{1}_3+A^{1}_1$
$A^{12}_1$	(0, 1, 0, 0, 0, 1, 0)	(4, 8, 10, 12, 14, 16, 8)	$3V_{4\psi}+8V_{3\psi}+11V_{2\psi}+8V_{\psi}+9V_{0}$	9	$2A^{4}_1+A^{3}_1$	24	12	$A^{2}_2+4A^{1}_1$ , $A^{2}_2+A^{2}_1+2A^{1}_1$ , $A^{2}_2+2A^{2}_1$	$A^{2}_2+4A^{1}_1$ , $A^{2}_2+A^{2}_1+2A^{1}_1$ , $A^{2}_2+2A^{2}_1$
$A^{11}_1$	(2, 0, 1, 0, 0, 0, 0)	(6, 8, 10, 10, 10, 10, 5)	$V_{6\psi}+V_{4\psi}+8V_{3\psi}+3V_{2\psi}+8V_{\psi}+36V_{0}$	36	$C^{1}_4$	22	11	$B^{1}_2+A^{1}_1$	$B^{1}_2+A^{1}_1$ , $C^{1}_3$
$A^{11}_1$	(0, 1, 0, 0, 1, 0, 0)	(4, 8, 10, 12, 14, 14, 7)	$3V_{4\psi}+6V_{3\psi}+11V_{2\psi}+12V_{\psi}+9V_{0}$	9	$A^{8}_1+A^{3}_1+A^{1}_1$	22	11	$A^{2}_2+3A^{1}_1$ , $A^{2}_2+A^{2}_1+A^{1}_1$	$A^{2}_2+3A^{1}_1$ , $A^{2}_2+A^{2}_1+A^{1}_1$
$A^{10}_1$	(2, 1, 0, 0, 0, 0, 0)	(6, 8, 8, 8, 8, 8, 4)	$V_{6\psi}+10V_{3\psi}+V_{2\psi}+55V_{0}$	55	$C^{1}_5$	20	10	$B^{1}_2$	$B^{1}_2$
$A^{10}_1$	(0, 1, 0, 1, 0, 0, 0)	(4, 8, 10, 12, 12, 12, 6)	$3V_{4\psi}+4V_{3\psi}+12V_{2\psi}+12V_{\psi}+14V_{0}$	14	$B^{1}_2+A^{3}_1$	20	10	$A^{2}_2+2A^{1}_1$ , $A^{2}_2+A^{2}_1$	$A^{2}_2+2A^{1}_1$ , $A^{2}_2+A^{2}_1$
$A^{9}_1$	(0, 1, 1, 0, 0, 0, 0)	(4, 8, 10, 10, 10, 10, 5)	$3V_{4\psi}+2V_{3\psi}+14V_{2\psi}+8V_{\psi}+24V_{0}$	24	$C^{1}_3+A^{3}_1$	18	9	$A^{2}_2+A^{1}_1$	$A^{2}_2+A^{1}_1$
$A^{8}_1$	(0, 2, 0, 0, 0, 0, 0)	(4, 8, 8, 8, 8, 8, 4)	$3V_{4\psi}+17V_{2\psi}+39V_{0}$	39	$C^{1}_4+A^{3}_1$	16	8	$A^{2}_2$	$A^{2}_2$
$A^{7}_1$	(0, 0, 0, 0, 0, 0, 2)	(2, 4, 6, 8, 10, 12, 7)	$28V_{2\psi}+21V_{0}$	21	$B^{4}_3$	14	7	$3A^{2}_1+A^{1}_1$ , $7A^{1}_1$ , $A^{2}_1+5A^{1}_1$ , $2A^{2}_1+3A^{1}_1$	$3A^{2}_1+A^{1}_1$ , $7A^{1}_1$ , $A^{2}_1+5A^{1}_1$ , $2A^{2}_1+3A^{1}_1$
$A^{6}_1$	(0, 0, 0, 0, 0, 1, 0)	(2, 4, 6, 8, 10, 12, 6)	$21V_{2\psi}+12V_{\psi}+18V_{0}$	18	$A^{4}_3+A^{1}_1$	12	6	$2A^{2}_1+2A^{1}_1$ , $3A^{2}_1$ , $6A^{1}_1$ , $A^{2}_1+4A^{1}_1$	$2A^{2}_1+2A^{1}_1$ , $3A^{2}_1$ , $6A^{1}_1$ , $A^{2}_1+4A^{1}_1$
$A^{5}_1$	(0, 0, 0, 0, 1, 0, 0)	(2, 4, 6, 8, 10, 10, 5)	$15V_{2\psi}+20V_{\psi}+20V_{0}$	20	$B^{4}_2+B^{1}_2$	10	5	$2A^{2}_1+A^{1}_1$ , $5A^{1}_1$ , $A^{2}_1+3A^{1}_1$	$2A^{2}_1+A^{1}_1$ , $5A^{1}_1$ , $A^{2}_1+3A^{1}_1$
$A^{4}_1$	(0, 0, 0, 1, 0, 0, 0)	(2, 4, 6, 8, 8, 8, 4)	$10V_{2\psi}+24V_{\psi}+27V_{0}$	27	$C^{1}_3+2A^{4}_1$	8	4	$A^{2}_1+2A^{1}_1$ , $2A^{2}_1$ , $4A^{1}_1$	$A^{2}_1+2A^{1}_1$ , $2A^{2}_1$ , $4A^{1}_1$
$A^{3}_1$	(0, 0, 1, 0, 0, 0, 0)	(2, 4, 6, 6, 6, 6, 3)	$6V_{2\psi}+24V_{\psi}+39V_{0}$	39	$C^{1}_4+A^{8}_1$	6	3	$A^{2}_1+A^{1}_1$ , $3A^{1}_1$	$A^{2}_1+A^{1}_1$ , $3A^{1}_1$
$A^{2}_1$	(0, 1, 0, 0, 0, 0, 0)	(2, 4, 4, 4, 4, 4, 2)	$3V_{2\psi}+20V_{\psi}+56V_{0}$	56	$C^{1}_5$	4	2	$2A^{1}_1$ , $A^{2}_1$	$2A^{1}_1$ , $A^{2}_1$
$A_1$	(1, 0, 0, 0, 0, 0, 0)	(2, 2, 2, 2, 2, 2, 1)	$V_{2\psi}+12V_{\psi}+78V_{0}$	78	$C^{1}_6$	2	1	$A^{1}_1$	$A^{1}_1$

Length longest root ambient algebra squared/4= 1/2

Given a root subsystem

P

, and a root sub-subsystem

{P}_{0}

, in (10.2) of Semisimple subalgebras of semisimple Lie algebras, E. Dynkin defines a numerical constant

e{}\left(P, {P}_{0}\right)

(which we call Dynkin epsilon).
In Theorem 10.3, Dynkin proves that if an

sl{}\left(2\right)

is an

S

-subalgebra in the root subalgebra generated by

P

, such that it has characteristic 2 for all simple roots of

P

lying in

{P}_{0}

, then

e{}\left(P, {P}_{0}\right)=0

H that wasn't realized on the first attempt but was ultimately realized: (22, 40, 54, 64, 70, 72, 37), Type: ${A^{287}}_{1}$ .
It turns out that in the current case of Cartan element h = (22, 40, 54, 64, 70, 72, 37) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

H that wasn't realized on the first attempt but was ultimately realized: (18, 32, 42, 48, 54, 56, 29), Type: ${A^{175}}_{1}$ .
It turns out that in the current case of Cartan element h = (18, 32, 42, 48, 54, 56, 29) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

H that wasn't realized on the first attempt but was ultimately realized: (18, 32, 42, 48, 50, 52, 27), Type: ${A^{167}}_{1}$ .
It turns out that in the current case of Cartan element h = (18, 32, 42, 48, 50, 52, 27) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

H that wasn't realized on the first attempt but was ultimately realized: (18, 32, 42, 48, 50, 52, 26), Type: ${A^{166}}_{1}$ .
It turns out that in the current case of Cartan element h = (18, 32, 42, 48, 50, 52, 26) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

H that wasn't realized on the first attempt but was ultimately realized: (14, 24, 34, 40, 46, 48, 25), Type: ${A^{119}}_{1}$ .
It turns out that in the current case of Cartan element h = (14, 24, 34, 40, 46, 48, 25) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

H that wasn't realized on the first attempt but was ultimately realized: (14, 24, 30, 36, 38, 40, 21), Type: ${A^{95}}_{1}$ .
It turns out that in the current case of Cartan element h = (14, 24, 30, 36, 38, 40, 21) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

H that wasn't realized on the first attempt but was ultimately realized: (14, 24, 30, 36, 38, 40, 20), Type: ${A^{94}}_{1}$ .
It turns out that in the current case of Cartan element h = (14, 24, 30, 36, 38, 40, 20) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

H that wasn't realized on the first attempt but was ultimately realized: (14, 24, 30, 32, 34, 36, 19), Type: ${A^{87}}_{1}$ .
It turns out that in the current case of Cartan element h = (14, 24, 30, 32, 34, 36, 19) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

H that wasn't realized on the first attempt but was ultimately realized: (14, 24, 30, 32, 34, 36, 18), Type: ${A^{86}}_{1}$ .
It turns out that in the current case of Cartan element h = (14, 24, 30, 32, 34, 36, 18) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

H that wasn't realized on the first attempt but was ultimately realized: (14, 24, 30, 32, 34, 34, 17), Type: ${A^{85}}_{1}$ .
It turns out that in the current case of Cartan element h = (14, 24, 30, 32, 34, 34, 17) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

H that wasn't realized on the first attempt but was ultimately realized: (10, 20, 26, 32, 34, 36, 19), Type: ${A^{71}}_{1}$ .
It turns out that in the current case of Cartan element h = (10, 20, 26, 32, 34, 36, 19) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

H that wasn't realized on the first attempt but was ultimately realized: (10, 16, 22, 28, 30, 32, 17), Type: ${A^{55}}_{1}$ .
It turns out that in the current case of Cartan element h = (10, 16, 22, 28, 30, 32, 17) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

H that wasn't realized on the first attempt but was ultimately realized: (10, 16, 22, 24, 26, 28, 15), Type: ${A^{47}}_{1}$ .
It turns out that in the current case of Cartan element h = (10, 16, 22, 24, 26, 28, 15) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

H that wasn't realized on the first attempt but was ultimately realized: (10, 16, 22, 24, 26, 28, 14), Type: ${A^{46}}_{1}$ .
It turns out that in the current case of Cartan element h = (10, 16, 22, 24, 26, 28, 14) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

H that wasn't realized on the first attempt but was ultimately realized: (10, 16, 22, 24, 26, 26, 13), Type: ${A^{45}}_{1}$ .
It turns out that in the current case of Cartan element h = (10, 16, 22, 24, 26, 26, 13) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

H that wasn't realized on the first attempt but was ultimately realized: (10, 16, 20, 24, 26, 28, 14), Type: ${A^{44}}_{1}$ .
It turns out that in the current case of Cartan element h = (10, 16, 20, 24, 26, 28, 14) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

H that wasn't realized on the first attempt but was ultimately realized: (10, 16, 18, 20, 22, 24, 13), Type: ${A^{39}}_{1}$ .
It turns out that in the current case of Cartan element h = (10, 16, 18, 20, 22, 24, 13) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

H that wasn't realized on the first attempt but was ultimately realized: (10, 16, 18, 20, 22, 24, 12), Type: ${A^{38}}_{1}$ .
It turns out that in the current case of Cartan element h = (10, 16, 18, 20, 22, 24, 12) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

H that wasn't realized on the first attempt but was ultimately realized: (10, 16, 18, 20, 22, 22, 11), Type: ${A^{37}}_{1}$ .
It turns out that in the current case of Cartan element h = (10, 16, 18, 20, 22, 22, 11) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

H that wasn't realized on the first attempt but was ultimately realized: (10, 16, 18, 20, 20, 20, 10), Type: ${A^{36}}_{1}$ .
It turns out that in the current case of Cartan element h = (10, 16, 18, 20, 20, 20, 10) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

H that wasn't realized on the first attempt but was ultimately realized: (6, 12, 18, 20, 22, 24, 13), Type: ${A^{31}}_{1}$ .
It turns out that in the current case of Cartan element h = (6, 12, 18, 20, 22, 24, 13) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

H that wasn't realized on the first attempt but was ultimately realized: (6, 12, 14, 16, 18, 20, 11), Type: ${A^{23}}_{1}$ .
It turns out that in the current case of Cartan element h = (6, 12, 14, 16, 18, 20, 11) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

H that wasn't realized on the first attempt but was ultimately realized: (6, 12, 14, 16, 18, 20, 10), Type: ${A^{22}}_{1}$ .
It turns out that in the current case of Cartan element h = (6, 12, 14, 16, 18, 20, 10) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

H that wasn't realized on the first attempt but was ultimately realized: (6, 12, 14, 16, 18, 18, 9), Type: ${A^{21}}_{1}$ .
It turns out that in the current case of Cartan element h = (6, 12, 14, 16, 18, 18, 9) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

H that wasn't realized on the first attempt but was ultimately realized: (6, 10, 14, 16, 18, 20, 10), Type: ${A^{20}}_{1}$ .
It turns out that in the current case of Cartan element h = (6, 10, 14, 16, 18, 20, 10) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

H that wasn't realized on the first attempt but was ultimately realized: (6, 10, 14, 16, 18, 18, 9), Type: ${A^{19}}_{1}$ .
It turns out that in the current case of Cartan element h = (6, 10, 14, 16, 18, 18, 9) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

H that wasn't realized on the first attempt but was ultimately realized: (6, 8, 10, 12, 14, 16, 9), Type: ${A^{15}}_{1}$ .
It turns out that in the current case of Cartan element h = (6, 8, 10, 12, 14, 16, 9) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

H that wasn't realized on the first attempt but was ultimately realized: (6, 8, 10, 12, 14, 16, 8), Type: ${A^{14}}_{1}$ .
It turns out that in the current case of Cartan element h = (6, 8, 10, 12, 14, 16, 8) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

H that wasn't realized on the first attempt but was ultimately realized: (6, 8, 10, 12, 14, 14, 7), Type: ${A^{13}}_{1}$ .
It turns out that in the current case of Cartan element h = (6, 8, 10, 12, 14, 14, 7) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

H that wasn't realized on the first attempt but was ultimately realized: (6, 8, 10, 12, 12, 12, 6), Type: ${A^{12}}_{1}$ .
It turns out that in the current case of Cartan element h = (6, 8, 10, 12, 12, 12, 6) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

H that wasn't realized on the first attempt but was ultimately realized: (6, 8, 10, 10, 10, 10, 5), Type: ${A^{11}}_{1}$ .
It turns out that in the current case of Cartan element h = (6, 8, 10, 10, 10, 10, 5) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

H that wasn't realized on the first attempt but was ultimately realized: (2, 4, 6, 8, 10, 12, 7), Type: ${A^{7}}_{1}$ .
It turns out that in the current case of Cartan element h = (2, 4, 6, 8, 10, 12, 7) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

H that wasn't realized on the first attempt but was ultimately realized: (2, 4, 6, 8, 10, 12, 6), Type: ${A^{6}}_{1}$ .
It turns out that in the current case of Cartan element h = (2, 4, 6, 8, 10, 12, 6) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

H that wasn't realized on the first attempt but was ultimately realized: (2, 4, 6, 8, 10, 10, 5), Type: ${A^{5}}_{1}$ .
It turns out that in the current case of Cartan element h = (2, 4, 6, 8, 10, 10, 5) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

H that wasn't realized on the first attempt but was ultimately realized: (2, 4, 6, 8, 8, 8, 4), Type: ${A^{4}}_{1}$ .
It turns out that in the current case of Cartan element h = (2, 4, 6, 8, 8, 8, 4) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

H that wasn't realized on the first attempt but was ultimately realized: (2, 4, 6, 6, 6, 6, 3), Type: ${A^{3}}_{1}$ .
It turns out that in the current case of Cartan element h = (2, 4, 6, 6, 6, 6, 3) we have that, for a certain P, e(P, P_0) equals 0, but I failed to realize the corresponding sl(2) as a subalgebra of that P. However, it turns out that h is indeed an S-subalgebra of a smaller root subalgebra P'.

Extensions of the rationals used (6 total):

\mathbb Q

\mathbb Q[\sqrt{-1}, \sqrt{3}]

\mathbb Q[\sqrt{-1}, \sqrt{2}]

\mathbb Q[\sqrt{-1}, \sqrt{2}, \sqrt{3}]

\mathbb Q[\sqrt{-1}]

\mathbb Q[\sqrt{-1}, \sqrt{3}, \sqrt{5}]

A^{455}_1

h-characteristic: (2, 2, 2, 2, 2, 2, 2)
Length of the weight dual to h: 910
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1:

C^{1}_7

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

V_{26\psi}+V_{22\psi}+V_{18\psi}+V_{14\psi}+V_{10\psi}+V_{6\psi}+V_{2\psi}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&49h_{7}+96h_{6}+90h_{5}+80h_{4}+66h_{3}+48h_{2}+26h_{1}\\ e&=&49g_{7}+48g_{6}+45g_{5}+40g_{4}+33g_{3}+24g_{2}+13g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-3}+g_{-4}+g_{-5}+g_{-6}+g_{-7}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&49h_{7}+96h_{6}+90h_{5}+80h_{4}+66h_{3}+48h_{2}+26h_{1}\\ [h, e]&=&98g_{7}+96g_{6}+90g_{5}+80g_{4}+66g_{3}+48g_{2}+26g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-3}-2g_{-4}-2g_{-5}-2g_{-6}-2g_{-7}\end{array}

Centralizer type:

0

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&49h_{7}+96h_{6}+90h_{5}+80h_{4}+66h_{3}+48h_{2}+26h_{1}\\ e&=&x_{7} g_{7}+x_{6} g_{6}+x_{5} g_{5}+x_{4} g_{4}+x_{3} g_{3}+x_{2} g_{2}+x_{1} g_{1}\\ f&=&x_{8} g_{-1}+x_{9} g_{-2}+x_{10} g_{-3}+x_{11} g_{-4}+x_{12} g_{-5}+x_{13} g_{-6}+x_{14} g_{-7}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(x_{7} x_{14} -49\right)h_{7}+\left(2x_{6} x_{13} -96\right)h_{6}+\left(2x_{5} x_{12} -90\right)h_{5}+\left(2x_{4} x_{11} -80\right)h_{4}+\left(2x_{3} x_{10} -66\right)h_{3}+\left(2x_{2} x_{9} -48\right)h_{2}+\left(2x_{1} x_{8} -26\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{8} -26&=&0\\2x_{2} x_{9} -48&=&0\\2x_{3} x_{10} -66&=&0\\2x_{4} x_{11} -80&=&0\\2x_{5} x_{12} -90&=&0\\2x_{6} x_{13} -96&=&0\\x_{7} x_{14} -49&=&0\\\end{array}

Starting h, e, f triple. H is computed according to Dynkin, and the coefficients of f are arbitrarily chosen.
More precisely, the chevalley generators participating in f are ordered in the order in which their roots appear, and the coefficients are chosen arbitrarily. More precisely, the n^th coefficient either 1) equals (n-1)^2+1 or 2) equals a hard-coded number that is specific to the given ambient Lie algebra, dynkin index and h element. Whenever a hard-coded coefficient is used, it was selected so it results in fast computations. The selection was discovered through manual experimentation. As of writing, the arbitrary coefficient selection happens here.

\begin{array}{rcl}h&=&49h_{7}+96h_{6}+90h_{5}+80h_{4}+66h_{3}+48h_{2}+26h_{1}\\e&=&x_{7} g_{7}+x_{6} g_{6}+x_{5} g_{5}+x_{4} g_{4}+x_{3} g_{3}+x_{2} g_{2}+x_{1} g_{1}\\f&=&g_{-1}+g_{-2}+g_{-3}+g_{-4}+g_{-5}+g_{-6}+g_{-7}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}26\\ 48\\ 66\\ 80\\ 90\\ 96\\ 49\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&49h_{7}+96h_{6}+90h_{5}+80h_{4}+66h_{3}+48h_{2}+26h_{1}\\ e&=&x_{7} g_{7}+x_{6} g_{6}+x_{5} g_{5}+x_{4} g_{4}+x_{3} g_{3}+x_{2} g_{2}+x_{1} g_{1}\\ f&=&x_{8} g_{-1}+x_{9} g_{-2}+x_{10} g_{-3}+x_{11} g_{-4}+x_{12} g_{-5}+x_{13} g_{-6}+x_{14} g_{-7}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{8} -26&=&0\\2x_{2} x_{9} -48&=&0\\2x_{3} x_{10} -66&=&0\\2x_{4} x_{11} -80&=&0\\2x_{5} x_{12} -90&=&0\\2x_{6} x_{13} -96&=&0\\x_{7} x_{14} -49&=&0\\\end{array}

A^{287}_1

h-characteristic: (2, 2, 2, 2, 2, 0, 2)
Length of the weight dual to h: 574
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1:

C^{1}_6+A^{1}_1

Containing regular semisimple subalgebra number 2:

C^{1}_7

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

V_{22\psi}+V_{18\psi}+V_{14\psi}+V_{12\psi}+2V_{10\psi}+V_{6\psi}+2V_{2\psi}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&37h_{7}+72h_{6}+70h_{5}+64h_{4}+54h_{3}+40h_{2}+22h_{1}\\ e&=&g_{19}+35g_{12}+36g_{7}+32g_{4}+27g_{3}+20g_{2}+11g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-3}+g_{-4}+g_{-7}+g_{-12}+g_{-19}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&37h_{7}+72h_{6}+70h_{5}+64h_{4}+54h_{3}+40h_{2}+22h_{1}\\ [h, e]&=&2g_{19}+70g_{12}+72g_{7}+64g_{4}+54g_{3}+40g_{2}+22g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-3}-2g_{-4}-2g_{-7}-2g_{-12}-2g_{-19}\end{array}

Centralizer type:

0

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&37h_{7}+72h_{6}+70h_{5}+64h_{4}+54h_{3}+40h_{2}+22h_{1}\\ e&=&x_{7} g_{19}+x_{5} g_{12}+x_{6} g_{7}+x_{4} g_{4}+x_{3} g_{3}+x_{2} g_{2}+x_{1} g_{1}\\ f&=&x_{8} g_{-1}+x_{9} g_{-2}+x_{10} g_{-3}+x_{11} g_{-4}+x_{13} g_{-7}+x_{12} g_{-12}+x_{14} g_{-19}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(x_{6} x_{13} +x_{7} x_{14} -37\right)h_{7}+\left(2x_{5} x_{12} +2x_{7} x_{14} -72\right)h_{6}+\left(2x_{5} x_{12} -70\right)h_{5}+\left(2x_{4} x_{11} -64\right)h_{4}+\left(2x_{3} x_{10} -54\right)h_{3}+\left(2x_{2} x_{9} -40\right)h_{2}+\left(2x_{1} x_{8} -22\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{8} -22&=&0\\2x_{2} x_{9} -40&=&0\\2x_{3} x_{10} -54&=&0\\2x_{4} x_{11} -64&=&0\\2x_{5} x_{12} -70&=&0\\2x_{5} x_{12} +2x_{7} x_{14} -72&=&0\\x_{6} x_{13} +x_{7} x_{14} -37&=&0\\\end{array}

\begin{array}{rcl}h&=&37h_{7}+72h_{6}+70h_{5}+64h_{4}+54h_{3}+40h_{2}+22h_{1}\\e&=&x_{7} g_{19}+x_{5} g_{12}+x_{6} g_{7}+x_{4} g_{4}+x_{3} g_{3}+x_{2} g_{2}+x_{1} g_{1}\\f&=&g_{-1}+g_{-2}+g_{-3}+g_{-4}+g_{-7}+g_{-12}+g_{-19}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 0 & 2 & 0 & 2\\ 0 & 0 & 0 & 0 & 0 & 1 & 1\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}22\\ 40\\ 54\\ 64\\ 70\\ 72\\ 37\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&37h_{7}+72h_{6}+70h_{5}+64h_{4}+54h_{3}+40h_{2}+22h_{1}\\ e&=&x_{7} g_{19}+x_{5} g_{12}+x_{6} g_{7}+x_{4} g_{4}+x_{3} g_{3}+x_{2} g_{2}+x_{1} g_{1}\\ f&=&x_{8} g_{-1}+x_{9} g_{-2}+x_{10} g_{-3}+x_{11} g_{-4}+x_{13} g_{-7}+x_{12} g_{-12}+x_{14} g_{-19}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{8} -22&=&0\\2x_{2} x_{9} -40&=&0\\2x_{3} x_{10} -54&=&0\\2x_{4} x_{11} -64&=&0\\2x_{5} x_{12} -70&=&0\\2x_{5} x_{12} +2x_{7} x_{14} -72&=&0\\x_{6} x_{13} +x_{7} x_{14} -37&=&0\\\end{array}

A^{286}_1

h-characteristic: (2, 2, 2, 2, 2, 1, 0)
Length of the weight dual to h: 572
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1:

C^{1}_6

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

V_{22\psi}+V_{18\psi}+V_{14\psi}+2V_{11\psi}+V_{10\psi}+V_{6\psi}+V_{2\psi}+3V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&36h_{7}+72h_{6}+70h_{5}+64h_{4}+54h_{3}+40h_{2}+22h_{1}\\ e&=&36g_{19}+35g_{5}+32g_{4}+27g_{3}+20g_{2}+11g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-3}+g_{-4}+g_{-5}+g_{-19}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&36h_{7}+72h_{6}+70h_{5}+64h_{4}+54h_{3}+40h_{2}+22h_{1}\\ [h, e]&=&72g_{19}+70g_{5}+64g_{4}+54g_{3}+40g_{2}+22g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-3}-2g_{-4}-2g_{-5}-2g_{-19}\end{array}

Centralizer type:

A_1

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&36h_{7}+72h_{6}+70h_{5}+64h_{4}+54h_{3}+40h_{2}+22h_{1}\\ e&=&x_{6} g_{19}+x_{5} g_{5}+x_{4} g_{4}+x_{3} g_{3}+x_{2} g_{2}+x_{1} g_{1}\\ f&=&x_{7} g_{-1}+x_{8} g_{-2}+x_{9} g_{-3}+x_{10} g_{-4}+x_{11} g_{-5}+x_{12} g_{-19}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(x_{6} x_{12} -36\right)h_{7}+\left(2x_{6} x_{12} -72\right)h_{6}+\left(2x_{5} x_{11} -70\right)h_{5}+\left(2x_{4} x_{10} -64\right)h_{4}+\left(2x_{3} x_{9} -54\right)h_{3}+\left(2x_{2} x_{8} -40\right)h_{2}+\left(2x_{1} x_{7} -22\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{7} -22&=&0\\2x_{2} x_{8} -40&=&0\\2x_{3} x_{9} -54&=&0\\2x_{4} x_{10} -64&=&0\\2x_{5} x_{11} -70&=&0\\2x_{6} x_{12} -72&=&0\\x_{6} x_{12} -36&=&0\\\end{array}

\begin{array}{rcl}h&=&36h_{7}+72h_{6}+70h_{5}+64h_{4}+54h_{3}+40h_{2}+22h_{1}\\e&=&x_{6} g_{19}+x_{5} g_{5}+x_{4} g_{4}+x_{3} g_{3}+x_{2} g_{2}+x_{1} g_{1}\\f&=&g_{-1}+g_{-2}+g_{-3}+g_{-4}+g_{-5}+g_{-19}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 2\\ 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}22\\ 40\\ 54\\ 64\\ 70\\ 72\\ 36\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&36h_{7}+72h_{6}+70h_{5}+64h_{4}+54h_{3}+40h_{2}+22h_{1}\\ e&=&x_{6} g_{19}+x_{5} g_{5}+x_{4} g_{4}+x_{3} g_{3}+x_{2} g_{2}+x_{1} g_{1}\\ f&=&x_{7} g_{-1}+x_{8} g_{-2}+x_{9} g_{-3}+x_{10} g_{-4}+x_{11} g_{-5}+x_{12} g_{-19}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{7} -22&=&0\\2x_{2} x_{8} -40&=&0\\2x_{3} x_{9} -54&=&0\\2x_{4} x_{10} -64&=&0\\2x_{5} x_{11} -70&=&0\\2x_{6} x_{12} -72&=&0\\x_{6} x_{12} -36&=&0\\\end{array}

A^{175}_1

h-characteristic: (2, 2, 2, 0, 2, 0, 2)
Length of the weight dual to h: 350
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1:

C^{1}_5+B^{1}_2

Containing regular semisimple subalgebra number 2:

C^{1}_7

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

V_{18\psi}+V_{14\psi}+V_{12\psi}+2V_{10\psi}+V_{8\psi}+3V_{6\psi}+2V_{2\psi}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&29h_{7}+56h_{6}+54h_{5}+48h_{4}+42h_{3}+32h_{2}+18h_{1}\\ e&=&4g_{19}+24g_{12}+3g_{11}+21g_{10}+25g_{7}+16g_{2}+9g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-7}+g_{-10}+g_{-11}+g_{-12}+g_{-19}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&29h_{7}+56h_{6}+54h_{5}+48h_{4}+42h_{3}+32h_{2}+18h_{1}\\ [h, e]&=&8g_{19}+48g_{12}+6g_{11}+42g_{10}+50g_{7}+32g_{2}+18g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-7}-2g_{-10}-2g_{-11}-2g_{-12}-2g_{-19}\end{array}

Centralizer type:

0

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&29h_{7}+56h_{6}+54h_{5}+48h_{4}+42h_{3}+32h_{2}+18h_{1}\\ e&=&x_{6} g_{19}+x_{4} g_{12}+x_{7} g_{11}+x_{3} g_{10}+x_{5} g_{7}+x_{2} g_{2}+x_{1} g_{1}\\ f&=&x_{8} g_{-1}+x_{9} g_{-2}+x_{12} g_{-7}+x_{10} g_{-10}+x_{14} g_{-11}+x_{11} g_{-12}+x_{13} g_{-19}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(x_{5} x_{12} +x_{6} x_{13} -29\right)h_{7}+\left(2x_{4} x_{11} +2x_{6} x_{13} -56\right)h_{6}+\left(2x_{4} x_{11} +2x_{7} x_{14} -54\right)h_{5}+\left(2x_{3} x_{10} +2x_{7} x_{14} -48\right)h_{4}+\left(2x_{3} x_{10} -42\right)h_{3}+\left(2x_{2} x_{9} -32\right)h_{2}+\left(2x_{1} x_{8} -18\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{8} -18&=&0\\2x_{2} x_{9} -32&=&0\\2x_{3} x_{10} -42&=&0\\2x_{3} x_{10} +2x_{7} x_{14} -48&=&0\\2x_{4} x_{11} +2x_{7} x_{14} -54&=&0\\2x_{4} x_{11} +2x_{6} x_{13} -56&=&0\\x_{5} x_{12} +x_{6} x_{13} -29&=&0\\\end{array}

\begin{array}{rcl}h&=&29h_{7}+56h_{6}+54h_{5}+48h_{4}+42h_{3}+32h_{2}+18h_{1}\\e&=&x_{6} g_{19}+x_{4} g_{12}+x_{7} g_{11}+x_{3} g_{10}+x_{5} g_{7}+x_{2} g_{2}+x_{1} g_{1}\\f&=&g_{-1}+g_{-2}+g_{-7}+g_{-10}+g_{-11}+g_{-12}+g_{-19}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 2\\ 0 & 0 & 0 & 2 & 0 & 0 & 2\\ 0 & 0 & 0 & 2 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 1 & 1 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}18\\ 32\\ 42\\ 48\\ 54\\ 56\\ 29\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&29h_{7}+56h_{6}+54h_{5}+48h_{4}+42h_{3}+32h_{2}+18h_{1}\\ e&=&x_{6} g_{19}+x_{4} g_{12}+x_{7} g_{11}+x_{3} g_{10}+x_{5} g_{7}+x_{2} g_{2}+x_{1} g_{1}\\ f&=&x_{8} g_{-1}+x_{9} g_{-2}+x_{12} g_{-7}+x_{10} g_{-10}+x_{14} g_{-11}+x_{11} g_{-12}+x_{13} g_{-19}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{8} -18&=&0\\2x_{2} x_{9} -32&=&0\\2x_{3} x_{10} -42&=&0\\2x_{3} x_{10} +2x_{7} x_{14} -48&=&0\\2x_{4} x_{11} +2x_{7} x_{14} -54&=&0\\2x_{4} x_{11} +2x_{6} x_{13} -56&=&0\\x_{5} x_{12} +x_{6} x_{13} -29&=&0\\\end{array}

A^{167}_1

h-characteristic: (2, 2, 2, 2, 0, 0, 2)
Length of the weight dual to h: 334
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1:

C^{1}_5+2A^{1}_1

Containing regular semisimple subalgebra number 2:

C^{1}_5+A^{2}_1

Containing regular semisimple subalgebra number 3:

C^{1}_6+A^{1}_1

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

V_{18\psi}+V_{14\psi}+3V_{10\psi}+2V_{8\psi}+V_{6\psi}+4V_{2\psi}+V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&27h_{7}+52h_{6}+50h_{5}+48h_{4}+42h_{3}+32h_{2}+18h_{1}\\ e&=&g_{29}+g_{19}+24g_{17}+25g_{7}+21g_{3}+16g_{2}+9g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-3}+g_{-7}+g_{-17}+g_{-19}+g_{-29}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&27h_{7}+52h_{6}+50h_{5}+48h_{4}+42h_{3}+32h_{2}+18h_{1}\\ [h, e]&=&2g_{29}+2g_{19}+48g_{17}+50g_{7}+42g_{3}+32g_{2}+18g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-3}-2g_{-7}-2g_{-17}-2g_{-19}-2g_{-29}\end{array}

Centralizer type:

0

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&27h_{7}+52h_{6}+50h_{5}+48h_{4}+42h_{3}+32h_{2}+18h_{1}\\ e&=&x_{6} g_{29}+x_{7} g_{19}+x_{4} g_{17}+x_{5} g_{7}+x_{3} g_{3}+x_{2} g_{2}+x_{1} g_{1}\\ f&=&x_{8} g_{-1}+x_{9} g_{-2}+x_{10} g_{-3}+x_{12} g_{-7}+x_{11} g_{-17}+x_{14} g_{-19}+x_{13} g_{-29}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -27\right)h_{7}+\left(2x_{4} x_{11} +2x_{6} x_{13} +2x_{7} x_{14} -52\right)h_{6}+\left(2x_{4} x_{11} +2x_{6} x_{13} -50\right)h_{5}+\left(2x_{4} x_{11} -48\right)h_{4}+\left(2x_{3} x_{10} -42\right)h_{3}+\left(2x_{2} x_{9} -32\right)h_{2}+\left(2x_{1} x_{8} -18\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{8} -18&=&0\\2x_{2} x_{9} -32&=&0\\2x_{3} x_{10} -42&=&0\\2x_{4} x_{11} -48&=&0\\2x_{4} x_{11} +2x_{6} x_{13} -50&=&0\\2x_{4} x_{11} +2x_{6} x_{13} +2x_{7} x_{14} -52&=&0\\x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -27&=&0\\\end{array}

\begin{array}{rcl}h&=&27h_{7}+52h_{6}+50h_{5}+48h_{4}+42h_{3}+32h_{2}+18h_{1}\\e&=&x_{6} g_{29}+x_{7} g_{19}+x_{4} g_{17}+x_{5} g_{7}+x_{3} g_{3}+x_{2} g_{2}+x_{1} g_{1}\\f&=&g_{-1}+g_{-2}+g_{-3}+g_{-7}+g_{-17}+g_{-19}+g_{-29}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 2 & 0\\ 0 & 0 & 0 & 2 & 0 & 2 & 2\\ 0 & 0 & 0 & 0 & 1 & 1 & 1\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}18\\ 32\\ 42\\ 48\\ 50\\ 52\\ 27\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&27h_{7}+52h_{6}+50h_{5}+48h_{4}+42h_{3}+32h_{2}+18h_{1}\\ e&=&x_{6} g_{29}+x_{7} g_{19}+x_{4} g_{17}+x_{5} g_{7}+x_{3} g_{3}+x_{2} g_{2}+x_{1} g_{1}\\ f&=&x_{8} g_{-1}+x_{9} g_{-2}+x_{10} g_{-3}+x_{12} g_{-7}+x_{11} g_{-17}+x_{14} g_{-19}+x_{13} g_{-29}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{8} -18&=&0\\2x_{2} x_{9} -32&=&0\\2x_{3} x_{10} -42&=&0\\2x_{4} x_{11} -48&=&0\\2x_{4} x_{11} +2x_{6} x_{13} -50&=&0\\2x_{4} x_{11} +2x_{6} x_{13} +2x_{7} x_{14} -52&=&0\\x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -27&=&0\\\end{array}

A^{166}_1

h-characteristic: (2, 2, 2, 2, 0, 1, 0)
Length of the weight dual to h: 332
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1:

C^{1}_5+A^{1}_1

Containing regular semisimple subalgebra number 2:

C^{1}_6

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

V_{18\psi}+V_{14\psi}+2V_{10\psi}+2V_{9\psi}+V_{8\psi}+V_{6\psi}+2V_{2\psi}+2V_{\psi}+3V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&26h_{7}+52h_{6}+50h_{5}+48h_{4}+42h_{3}+32h_{2}+18h_{1}\\ e&=&g_{29}+25g_{19}+24g_{11}+21g_{3}+16g_{2}+9g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-3}+g_{-11}+g_{-19}+g_{-29}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&26h_{7}+52h_{6}+50h_{5}+48h_{4}+42h_{3}+32h_{2}+18h_{1}\\ [h, e]&=&2g_{29}+50g_{19}+48g_{11}+42g_{3}+32g_{2}+18g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-3}-2g_{-11}-2g_{-19}-2g_{-29}\end{array}

Centralizer type:

A_1

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&26h_{7}+52h_{6}+50h_{5}+48h_{4}+42h_{3}+32h_{2}+18h_{1}\\ e&=&x_{6} g_{29}+x_{5} g_{19}+x_{4} g_{11}+x_{3} g_{3}+x_{2} g_{2}+x_{1} g_{1}\\ f&=&x_{7} g_{-1}+x_{8} g_{-2}+x_{9} g_{-3}+x_{10} g_{-11}+x_{11} g_{-19}+x_{12} g_{-29}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(x_{5} x_{11} +x_{6} x_{12} -26\right)h_{7}+\left(2x_{5} x_{11} +2x_{6} x_{12} -52\right)h_{6}+\left(2x_{4} x_{10} +2x_{6} x_{12} -50\right)h_{5}+\left(2x_{4} x_{10} -48\right)h_{4}+\left(2x_{3} x_{9} -42\right)h_{3}+\left(2x_{2} x_{8} -32\right)h_{2}+\left(2x_{1} x_{7} -18\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{7} -18&=&0\\2x_{2} x_{8} -32&=&0\\2x_{3} x_{9} -42&=&0\\2x_{4} x_{10} -48&=&0\\2x_{4} x_{10} +2x_{6} x_{12} -50&=&0\\2x_{5} x_{11} +2x_{6} x_{12} -52&=&0\\x_{5} x_{11} +x_{6} x_{12} -26&=&0\\\end{array}

\begin{array}{rcl}h&=&26h_{7}+52h_{6}+50h_{5}+48h_{4}+42h_{3}+32h_{2}+18h_{1}\\e&=&x_{6} g_{29}+x_{5} g_{19}+x_{4} g_{11}+x_{3} g_{3}+x_{2} g_{2}+x_{1} g_{1}\\f&=&g_{-1}+g_{-2}+g_{-3}+g_{-11}+g_{-19}+g_{-29}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 2\\ 0 & 0 & 0 & 0 & 2 & 2\\ 0 & 0 & 0 & 0 & 1 & 1\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}18\\ 32\\ 42\\ 48\\ 50\\ 52\\ 26\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&26h_{7}+52h_{6}+50h_{5}+48h_{4}+42h_{3}+32h_{2}+18h_{1}\\ e&=&x_{6} g_{29}+x_{5} g_{19}+x_{4} g_{11}+x_{3} g_{3}+x_{2} g_{2}+x_{1} g_{1}\\ f&=&x_{7} g_{-1}+x_{8} g_{-2}+x_{9} g_{-3}+x_{10} g_{-11}+x_{11} g_{-19}+x_{12} g_{-29}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{7} -18&=&0\\2x_{2} x_{8} -32&=&0\\2x_{3} x_{9} -42&=&0\\2x_{4} x_{10} -48&=&0\\2x_{4} x_{10} +2x_{6} x_{12} -50&=&0\\2x_{5} x_{11} +2x_{6} x_{12} -52&=&0\\x_{5} x_{11} +x_{6} x_{12} -26&=&0\\\end{array}

A^{165}_1

h-characteristic: (2, 2, 2, 2, 1, 0, 0)
Length of the weight dual to h: 330
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1:

C^{1}_5

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

V_{18\psi}+V_{14\psi}+V_{10\psi}+4V_{9\psi}+V_{6\psi}+V_{2\psi}+10V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&25h_{7}+50h_{6}+50h_{5}+48h_{4}+42h_{3}+32h_{2}+18h_{1}\\ e&=&25g_{29}+24g_{4}+21g_{3}+16g_{2}+9g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-3}+g_{-4}+g_{-29}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&25h_{7}+50h_{6}+50h_{5}+48h_{4}+42h_{3}+32h_{2}+18h_{1}\\ [h, e]&=&50g_{29}+48g_{4}+42g_{3}+32g_{2}+18g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-3}-2g_{-4}-2g_{-29}\end{array}

Centralizer type:

B_2

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&25h_{7}+50h_{6}+50h_{5}+48h_{4}+42h_{3}+32h_{2}+18h_{1}\\ e&=&x_{5} g_{29}+x_{4} g_{4}+x_{3} g_{3}+x_{2} g_{2}+x_{1} g_{1}\\ f&=&x_{6} g_{-1}+x_{7} g_{-2}+x_{8} g_{-3}+x_{9} g_{-4}+x_{10} g_{-29}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(x_{5} x_{10} -25\right)h_{7}+\left(2x_{5} x_{10} -50\right)h_{6}+\left(2x_{5} x_{10} -50\right)h_{5}+\left(2x_{4} x_{9} -48\right)h_{4}+\left(2x_{3} x_{8} -42\right)h_{3}+\left(2x_{2} x_{7} -32\right)h_{2}+\left(2x_{1} x_{6} -18\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{6} -18&=&0\\2x_{2} x_{7} -32&=&0\\2x_{3} x_{8} -42&=&0\\2x_{4} x_{9} -48&=&0\\2x_{5} x_{10} -50&=&0\\2x_{5} x_{10} -50&=&0\\x_{5} x_{10} -25&=&0\\\end{array}

\begin{array}{rcl}h&=&25h_{7}+50h_{6}+50h_{5}+48h_{4}+42h_{3}+32h_{2}+18h_{1}\\e&=&x_{5} g_{29}+x_{4} g_{4}+x_{3} g_{3}+x_{2} g_{2}+x_{1} g_{1}\\f&=&g_{-1}+g_{-2}+g_{-3}+g_{-4}+g_{-29}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 2\\ 0 & 0 & 0 & 0 & 2\\ 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}18\\ 32\\ 42\\ 48\\ 50\\ 50\\ 25\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&25h_{7}+50h_{6}+50h_{5}+48h_{4}+42h_{3}+32h_{2}+18h_{1}\\ e&=&x_{5} g_{29}+x_{4} g_{4}+x_{3} g_{3}+x_{2} g_{2}+x_{1} g_{1}\\ f&=&x_{6} g_{-1}+x_{7} g_{-2}+x_{8} g_{-3}+x_{9} g_{-4}+x_{10} g_{-29}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{6} -18&=&0\\2x_{2} x_{7} -32&=&0\\2x_{3} x_{8} -42&=&0\\2x_{4} x_{9} -48&=&0\\2x_{5} x_{10} -50&=&0\\2x_{5} x_{10} -50&=&0\\x_{5} x_{10} -25&=&0\\\end{array}

A^{119}_1

h-characteristic: (2, 0, 2, 0, 2, 0, 2)
Length of the weight dual to h: 238
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1:

C^{1}_4+C^{1}_3

Containing regular semisimple subalgebra number 2:

C^{1}_7

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

V_{14\psi}+V_{12\psi}+3V_{10\psi}+V_{8\psi}+3V_{6\psi}+V_{4\psi}+3V_{2\psi}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&25h_{7}+48h_{6}+46h_{5}+40h_{4}+34h_{3}+24h_{2}+14h_{1}\\ e&=&9g_{19}+15g_{12}+8g_{11}+12g_{10}+5g_{9}+7g_{8}+16g_{7}\\ f&=&g_{-7}+g_{-8}+g_{-9}+g_{-10}+g_{-11}+g_{-12}+g_{-19}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&25h_{7}+48h_{6}+46h_{5}+40h_{4}+34h_{3}+24h_{2}+14h_{1}\\ [h, e]&=&18g_{19}+30g_{12}+16g_{11}+24g_{10}+10g_{9}+14g_{8}+32g_{7}\\ [h, f]&=&-2g_{-7}-2g_{-8}-2g_{-9}-2g_{-10}-2g_{-11}-2g_{-12}-2g_{-19}\end{array}

Centralizer type:

0

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&25h_{7}+48h_{6}+46h_{5}+40h_{4}+34h_{3}+24h_{2}+14h_{1}\\ e&=&x_{7} g_{19}+x_{3} g_{12}+x_{6} g_{11}+x_{2} g_{10}+x_{5} g_{9}+x_{1} g_{8}+x_{4} g_{7}\\ f&=&x_{11} g_{-7}+x_{8} g_{-8}+x_{12} g_{-9}+x_{9} g_{-10}+x_{13} g_{-11}+x_{10} g_{-12}+x_{14} g_{-19}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(x_{4} x_{11} +x_{7} x_{14} -25\right)h_{7}+\left(2x_{3} x_{10} +2x_{7} x_{14} -48\right)h_{6}+\left(2x_{3} x_{10} +2x_{6} x_{13} -46\right)h_{5}+\left(2x_{2} x_{9} +2x_{6} x_{13} -40\right)h_{4}+\left(2x_{2} x_{9} +2x_{5} x_{12} -34\right)h_{3}+\left(2x_{1} x_{8} +2x_{5} x_{12} -24\right)h_{2}+\left(2x_{1} x_{8} -14\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{8} -14&=&0\\2x_{1} x_{8} +2x_{5} x_{12} -24&=&0\\2x_{2} x_{9} +2x_{5} x_{12} -34&=&0\\2x_{2} x_{9} +2x_{6} x_{13} -40&=&0\\2x_{3} x_{10} +2x_{6} x_{13} -46&=&0\\2x_{3} x_{10} +2x_{7} x_{14} -48&=&0\\x_{4} x_{11} +x_{7} x_{14} -25&=&0\\\end{array}

\begin{array}{rcl}h&=&25h_{7}+48h_{6}+46h_{5}+40h_{4}+34h_{3}+24h_{2}+14h_{1}\\e&=&x_{7} g_{19}+x_{3} g_{12}+x_{6} g_{11}+x_{2} g_{10}+x_{5} g_{9}+x_{1} g_{8}+x_{4} g_{7}\\f&=&g_{-7}+g_{-8}+g_{-9}+g_{-10}+g_{-11}+g_{-12}+g_{-19}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0\\ 2 & 0 & 0 & 0 & 2 & 0 & 0\\ 0 & 2 & 0 & 0 & 2 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 2 & 0 & 0 & 2 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 2\\ 0 & 0 & 0 & 1 & 0 & 0 & 1\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}14\\ 24\\ 34\\ 40\\ 46\\ 48\\ 25\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&25h_{7}+48h_{6}+46h_{5}+40h_{4}+34h_{3}+24h_{2}+14h_{1}\\ e&=&x_{7} g_{19}+x_{3} g_{12}+x_{6} g_{11}+x_{2} g_{10}+x_{5} g_{9}+x_{1} g_{8}+x_{4} g_{7}\\ f&=&x_{11} g_{-7}+x_{8} g_{-8}+x_{12} g_{-9}+x_{9} g_{-10}+x_{13} g_{-11}+x_{10} g_{-12}+x_{14} g_{-19}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{8} -14&=&0\\2x_{1} x_{8} +2x_{5} x_{12} -24&=&0\\2x_{2} x_{9} +2x_{5} x_{12} -34&=&0\\2x_{2} x_{9} +2x_{6} x_{13} -40&=&0\\2x_{3} x_{10} +2x_{6} x_{13} -46&=&0\\2x_{3} x_{10} +2x_{7} x_{14} -48&=&0\\x_{4} x_{11} +x_{7} x_{14} -25&=&0\\\end{array}

A^{112}_1

h-characteristic: (0, 2, 0, 2, 0, 2, 0)
Length of the weight dual to h: 224
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1:

A^{2}_6

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

3V_{12\psi}+V_{10\psi}+3V_{8\psi}+V_{6\psi}+3V_{4\psi}+V_{2\psi}+3V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&24h_{7}+48h_{6}+44h_{5}+40h_{4}+32h_{3}+24h_{2}+12h_{1}\\ e&=&6g_{14}+12g_{13}+12g_{12}+10g_{11}+10g_{10}+6g_{2}\\ f&=&g_{-2}+g_{-10}+g_{-11}+g_{-12}+g_{-13}+g_{-14}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&24h_{7}+48h_{6}+44h_{5}+40h_{4}+32h_{3}+24h_{2}+12h_{1}\\ [h, e]&=&12g_{14}+24g_{13}+24g_{12}+20g_{11}+20g_{10}+12g_{2}\\ [h, f]&=&-2g_{-2}-2g_{-10}-2g_{-11}-2g_{-12}-2g_{-13}-2g_{-14}\end{array}

Centralizer type:

A^{7}_1

Killing form square of Cartan element dual to ambient long root: 32
Basis of the centralizer (dimension: 3):

h_{7}+2h_{5}+2h_{3}-2h_{1}

g_{1}+g_{-3}+g_{-5}+g_{-7}

g_{7}+g_{5}+g_{3}+g_{-1}

Basis of centralizer intersected with cartan (dimension: 1):

1/2h_{7}+h_{5}+h_{3}-h_{1}

Cartan of centralizer (dimension: 1):

1/2h_{7}+h_{5}+h_{3}-h_{1}

Cartan-generating semisimple element:

1/2h_{7}+h_{5}+h_{3}-h_{1}

adjoint action:

\begin{pmatrix}0 & 0 & 0\\ 0 & -1 & 0\\ 0 & 0 & 1\\ \end{pmatrix}

Characteristic polynomial ad H:

x^3-x

Factorization of characteristic polynomial of ad H: (x )(x -1)(x +1)
Eigenvalues of ad H:

0

1

-1

3 eigenvectors of ad H:

(1,0,0)

(0,0,1)

(0,1,0)

Centralizer type: A^{7}_1
Reductive components (1 total):
Scalar product computed:

\begin{pmatrix}1/56\\ \end{pmatrix}

Simple basis of Cartan of centralizer (1 total):

h_{7}+2h_{5}+2h_{3}-2h_{1}

matching e:

g_{7}+g_{5}+g_{3}+g_{-1}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0\\ 0 & -2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}

Linear space basis of intersection of centralizer and ambient Cartan:

h_{7}+2h_{5}+2h_{3}-2h_{1}

matching e:

g_{7}+g_{5}+g_{3}+g_{-1}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0\\ 0 & -2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}

Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form:

\begin{pmatrix}224\\ \end{pmatrix}

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&24h_{7}+48h_{6}+44h_{5}+40h_{4}+32h_{3}+24h_{2}+12h_{1}\\ e&=&x_{1} g_{14}+x_{3} g_{13}+x_{4} g_{12}+x_{2} g_{11}+x_{5} g_{10}+x_{6} g_{2}\\ f&=&x_{12} g_{-2}+x_{11} g_{-10}+x_{8} g_{-11}+x_{10} g_{-12}+x_{9} g_{-13}+x_{7} g_{-14}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(2x_{3} x_{9} -24\right)h_{7}+\left(2x_{3} x_{9} +2x_{4} x_{10} -48\right)h_{6}+\left(2x_{2} x_{8} +2x_{4} x_{10} -44\right)h_{5}+\left(2x_{2} x_{8} +2x_{5} x_{11} -40\right)h_{4}+\left(2x_{1} x_{7} +2x_{5} x_{11} -32\right)h_{3}+\left(2x_{1} x_{7} +2x_{6} x_{12} -24\right)h_{2}+\left(2x_{1} x_{7} -12\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{7} -12&=&0\\2x_{1} x_{7} +2x_{6} x_{12} -24&=&0\\2x_{1} x_{7} +2x_{5} x_{11} -32&=&0\\2x_{2} x_{8} +2x_{5} x_{11} -40&=&0\\2x_{2} x_{8} +2x_{4} x_{10} -44&=&0\\2x_{3} x_{9} +2x_{4} x_{10} -48&=&0\\2x_{3} x_{9} -24&=&0\\\end{array}

\begin{array}{rcl}h&=&24h_{7}+48h_{6}+44h_{5}+40h_{4}+32h_{3}+24h_{2}+12h_{1}\\e&=&x_{1} g_{14}+x_{3} g_{13}+x_{4} g_{12}+x_{2} g_{11}+x_{5} g_{10}+x_{6} g_{2}\\f&=&g_{-2}+g_{-10}+g_{-11}+g_{-12}+g_{-13}+g_{-14}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0\\ 2 & 0 & 0 & 0 & 0 & 2\\ 2 & 0 & 0 & 0 & 2 & 0\\ 0 & 2 & 0 & 0 & 2 & 0\\ 0 & 2 & 0 & 2 & 0 & 0\\ 0 & 0 & 2 & 2 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}12\\ 24\\ 32\\ 40\\ 44\\ 48\\ 24\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&24h_{7}+48h_{6}+44h_{5}+40h_{4}+32h_{3}+24h_{2}+12h_{1}\\ e&=&x_{1} g_{14}+x_{3} g_{13}+x_{4} g_{12}+x_{2} g_{11}+x_{5} g_{10}+x_{6} g_{2}\\ f&=&x_{12} g_{-2}+x_{11} g_{-10}+x_{8} g_{-11}+x_{10} g_{-12}+x_{9} g_{-13}+x_{7} g_{-14}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{7} -12&=&0\\2x_{1} x_{7} +2x_{6} x_{12} -24&=&0\\2x_{1} x_{7} +2x_{5} x_{11} -32&=&0\\2x_{2} x_{8} +2x_{5} x_{11} -40&=&0\\2x_{2} x_{8} +2x_{4} x_{10} -44&=&0\\2x_{3} x_{9} +2x_{4} x_{10} -48&=&0\\2x_{3} x_{9} -24&=&0\\\end{array}

A^{95}_1

h-characteristic: (2, 2, 0, 2, 0, 0, 2)
Length of the weight dual to h: 190
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 5
Containing regular semisimple subalgebra number 1:

C^{1}_4+B^{1}_2+A^{1}_1

Containing regular semisimple subalgebra number 2:

C^{1}_7

Containing regular semisimple subalgebra number 3:

C^{1}_6+A^{1}_1

Containing regular semisimple subalgebra number 4:

C^{1}_5+B^{1}_2

Containing regular semisimple subalgebra number 5:

C^{1}_4+C^{1}_3

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

V_{14\psi}+2V_{10\psi}+2V_{8\psi}+4V_{6\psi}+2V_{4\psi}+4V_{2\psi}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&21h_{7}+40h_{6}+38h_{5}+36h_{4}+30h_{3}+24h_{2}+14h_{1}\\ e&=&4g_{29}+g_{19}+15g_{17}+3g_{10}+12g_{9}+16g_{7}+7g_{1}\\ f&=&g_{-1}+g_{-7}+g_{-9}+g_{-10}+g_{-17}+g_{-19}+g_{-29}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&21h_{7}+40h_{6}+38h_{5}+36h_{4}+30h_{3}+24h_{2}+14h_{1}\\ [h, e]&=&8g_{29}+2g_{19}+30g_{17}+6g_{10}+24g_{9}+32g_{7}+14g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-7}-2g_{-9}-2g_{-10}-2g_{-17}-2g_{-19}-2g_{-29}\end{array}

Centralizer type:

0

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&21h_{7}+40h_{6}+38h_{5}+36h_{4}+30h_{3}+24h_{2}+14h_{1}\\ e&=&x_{5} g_{29}+x_{7} g_{19}+x_{3} g_{17}+x_{6} g_{10}+x_{2} g_{9}+x_{4} g_{7}+x_{1} g_{1}\\ f&=&x_{8} g_{-1}+x_{11} g_{-7}+x_{9} g_{-9}+x_{13} g_{-10}+x_{10} g_{-17}+x_{14} g_{-19}+x_{12} g_{-29}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(x_{4} x_{11} +x_{5} x_{12} +x_{7} x_{14} -21\right)h_{7}+\left(2x_{3} x_{10} +2x_{5} x_{12} +2x_{7} x_{14} -40\right)h_{6}+\left(2x_{3} x_{10} +2x_{5} x_{12} -38\right)h_{5}+\left(2x_{3} x_{10} +2x_{6} x_{13} -36\right)h_{4}+\left(2x_{2} x_{9} +2x_{6} x_{13} -30\right)h_{3}+\left(2x_{2} x_{9} -24\right)h_{2}+\left(2x_{1} x_{8} -14\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{8} -14&=&0\\2x_{2} x_{9} -24&=&0\\2x_{2} x_{9} +2x_{6} x_{13} -30&=&0\\2x_{3} x_{10} +2x_{6} x_{13} -36&=&0\\2x_{3} x_{10} +2x_{5} x_{12} -38&=&0\\2x_{3} x_{10} +2x_{5} x_{12} +2x_{7} x_{14} -40&=&0\\x_{4} x_{11} +x_{5} x_{12} +x_{7} x_{14} -21&=&0\\\end{array}

\begin{array}{rcl}h&=&21h_{7}+40h_{6}+38h_{5}+36h_{4}+30h_{3}+24h_{2}+14h_{1}\\e&=&x_{5} g_{29}+x_{7} g_{19}+x_{3} g_{17}+x_{6} g_{10}+x_{2} g_{9}+x_{4} g_{7}+x_{1} g_{1}\\f&=&g_{-1}+g_{-7}+g_{-9}+g_{-10}+g_{-17}+g_{-19}+g_{-29}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 2 & 0 & 0 & 2 & 0\\ 0 & 0 & 2 & 0 & 2 & 0 & 0\\ 0 & 0 & 2 & 0 & 2 & 0 & 2\\ 0 & 0 & 0 & 1 & 1 & 0 & 1\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}14\\ 24\\ 30\\ 36\\ 38\\ 40\\ 21\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&21h_{7}+40h_{6}+38h_{5}+36h_{4}+30h_{3}+24h_{2}+14h_{1}\\ e&=&x_{5} g_{29}+x_{7} g_{19}+x_{3} g_{17}+x_{6} g_{10}+x_{2} g_{9}+x_{4} g_{7}+x_{1} g_{1}\\ f&=&x_{8} g_{-1}+x_{11} g_{-7}+x_{9} g_{-9}+x_{13} g_{-10}+x_{10} g_{-17}+x_{14} g_{-19}+x_{12} g_{-29}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{8} -14&=&0\\2x_{2} x_{9} -24&=&0\\2x_{2} x_{9} +2x_{6} x_{13} -30&=&0\\2x_{3} x_{10} +2x_{6} x_{13} -36&=&0\\2x_{3} x_{10} +2x_{5} x_{12} -38&=&0\\2x_{3} x_{10} +2x_{5} x_{12} +2x_{7} x_{14} -40&=&0\\x_{4} x_{11} +x_{5} x_{12} +x_{7} x_{14} -21&=&0\\\end{array}

A^{94}_1

h-characteristic: (2, 2, 0, 2, 0, 1, 0)
Length of the weight dual to h: 188
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1:

C^{1}_4+B^{1}_2

Containing regular semisimple subalgebra number 2:

C^{1}_6

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

V_{14\psi}+2V_{10\psi}+V_{8\psi}+2V_{7\psi}+3V_{6\psi}+V_{4\psi}+2V_{3\psi}+2V_{2\psi}+3V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&20h_{7}+40h_{6}+38h_{5}+36h_{4}+30h_{3}+24h_{2}+14h_{1}\\ e&=&4g_{29}+16g_{19}+15g_{11}+3g_{10}+12g_{9}+7g_{1}\\ f&=&g_{-1}+g_{-9}+g_{-10}+g_{-11}+g_{-19}+g_{-29}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&20h_{7}+40h_{6}+38h_{5}+36h_{4}+30h_{3}+24h_{2}+14h_{1}\\ [h, e]&=&8g_{29}+32g_{19}+30g_{11}+6g_{10}+24g_{9}+14g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-9}-2g_{-10}-2g_{-11}-2g_{-19}-2g_{-29}\end{array}

Centralizer type:

A_1

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&20h_{7}+40h_{6}+38h_{5}+36h_{4}+30h_{3}+24h_{2}+14h_{1}\\ e&=&x_{5} g_{29}+x_{4} g_{19}+x_{3} g_{11}+x_{6} g_{10}+x_{2} g_{9}+x_{1} g_{1}\\ f&=&x_{7} g_{-1}+x_{8} g_{-9}+x_{12} g_{-10}+x_{9} g_{-11}+x_{10} g_{-19}+x_{11} g_{-29}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(x_{4} x_{10} +x_{5} x_{11} -20\right)h_{7}+\left(2x_{4} x_{10} +2x_{5} x_{11} -40\right)h_{6}+\left(2x_{3} x_{9} +2x_{5} x_{11} -38\right)h_{5}+\left(2x_{3} x_{9} +2x_{6} x_{12} -36\right)h_{4}+\left(2x_{2} x_{8} +2x_{6} x_{12} -30\right)h_{3}+\left(2x_{2} x_{8} -24\right)h_{2}+\left(2x_{1} x_{7} -14\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{7} -14&=&0\\2x_{2} x_{8} -24&=&0\\2x_{2} x_{8} +2x_{6} x_{12} -30&=&0\\2x_{3} x_{9} +2x_{6} x_{12} -36&=&0\\2x_{3} x_{9} +2x_{5} x_{11} -38&=&0\\2x_{4} x_{10} +2x_{5} x_{11} -40&=&0\\x_{4} x_{10} +x_{5} x_{11} -20&=&0\\\end{array}

\begin{array}{rcl}h&=&20h_{7}+40h_{6}+38h_{5}+36h_{4}+30h_{3}+24h_{2}+14h_{1}\\e&=&x_{5} g_{29}+x_{4} g_{19}+x_{3} g_{11}+x_{6} g_{10}+x_{2} g_{9}+x_{1} g_{1}\\f&=&g_{-1}+g_{-9}+g_{-10}+g_{-11}+g_{-19}+g_{-29}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 2\\ 0 & 0 & 2 & 0 & 0 & 2\\ 0 & 0 & 2 & 0 & 2 & 0\\ 0 & 0 & 0 & 2 & 2 & 0\\ 0 & 0 & 0 & 1 & 1 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}14\\ 24\\ 30\\ 36\\ 38\\ 40\\ 20\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&20h_{7}+40h_{6}+38h_{5}+36h_{4}+30h_{3}+24h_{2}+14h_{1}\\ e&=&x_{5} g_{29}+x_{4} g_{19}+x_{3} g_{11}+x_{6} g_{10}+x_{2} g_{9}+x_{1} g_{1}\\ f&=&x_{7} g_{-1}+x_{8} g_{-9}+x_{12} g_{-10}+x_{9} g_{-11}+x_{10} g_{-19}+x_{11} g_{-29}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{7} -14&=&0\\2x_{2} x_{8} -24&=&0\\2x_{2} x_{8} +2x_{6} x_{12} -30&=&0\\2x_{3} x_{9} +2x_{6} x_{12} -36&=&0\\2x_{3} x_{9} +2x_{5} x_{11} -38&=&0\\2x_{4} x_{10} +2x_{5} x_{11} -40&=&0\\x_{4} x_{10} +x_{5} x_{11} -20&=&0\\\end{array}

A^{92}_1

h-characteristic: (2, 2, 1, 0, 1, 1, 0)
Length of the weight dual to h: 184
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1:

C^{1}_4+A^{2}_2

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

V_{14\psi}+V_{10\psi}+2V_{9\psi}+2V_{7\psi}+V_{6\psi}+2V_{5\psi}+3V_{4\psi}+2V_{2\psi}+3V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&20h_{7}+40h_{6}+38h_{5}+34h_{4}+30h_{3}+24h_{2}+14h_{1}\\ e&=&2g_{23}+16g_{19}+15g_{16}+2g_{12}+12g_{2}+7g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-12}+g_{-16}+g_{-19}+g_{-23}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&20h_{7}+40h_{6}+38h_{5}+34h_{4}+30h_{3}+24h_{2}+14h_{1}\\ [h, e]&=&4g_{23}+32g_{19}+30g_{16}+4g_{12}+24g_{2}+14g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-12}-2g_{-16}-2g_{-19}-2g_{-23}\end{array}

Centralizer type:

A^{3}_1

Killing form square of Cartan element dual to ambient long root: 32
Basis of the centralizer (dimension: 3):

h_{7}-2h_{4}

g_{4}+g_{-7}

g_{7}+g_{-4}

Basis of centralizer intersected with cartan (dimension: 1):

1/2h_{7}-h_{4}

Cartan of centralizer (dimension: 1):

1/2h_{7}-h_{4}

Cartan-generating semisimple element:

1/2h_{7}-h_{4}

adjoint action:

\begin{pmatrix}0 & 0 & 0\\ 0 & -1 & 0\\ 0 & 0 & 1\\ \end{pmatrix}

Characteristic polynomial ad H:

x^3-x

Factorization of characteristic polynomial of ad H: (x )(x -1)(x +1)
Eigenvalues of ad H:

0

1

-1

3 eigenvectors of ad H:

(1,0,0)

(0,0,1)

(0,1,0)

Centralizer type: A^{3}_1
Reductive components (1 total):
Scalar product computed:

\begin{pmatrix}1/24\\ \end{pmatrix}

Simple basis of Cartan of centralizer (1 total):

h_{7}-2h_{4}

matching e:

g_{7}+g_{-4}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0\\ 0 & -2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}

Linear space basis of intersection of centralizer and ambient Cartan:

h_{7}-2h_{4}

matching e:

g_{7}+g_{-4}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0\\ 0 & -2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}

Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form:

\begin{pmatrix}96\\ \end{pmatrix}

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&20h_{7}+40h_{6}+38h_{5}+34h_{4}+30h_{3}+24h_{2}+14h_{1}\\ e&=&x_{5} g_{23}+x_{4} g_{19}+x_{3} g_{16}+x_{6} g_{12}+x_{2} g_{2}+x_{1} g_{1}\\ f&=&x_{7} g_{-1}+x_{8} g_{-2}+x_{12} g_{-12}+x_{9} g_{-16}+x_{10} g_{-19}+x_{11} g_{-23}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(x_{4} x_{10} +2x_{5} x_{11} -20\right)h_{7}+\left(2x_{4} x_{10} +2x_{5} x_{11} +2x_{6} x_{12} -40\right)h_{6}+\left(2x_{3} x_{9} +2x_{5} x_{11} +2x_{6} x_{12} -38\right)h_{5}+\left(2x_{3} x_{9} +2x_{5} x_{11} -34\right)h_{4}+\left(2x_{3} x_{9} -30\right)h_{3}+\left(2x_{2} x_{8} -24\right)h_{2}+\left(2x_{1} x_{7} -14\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{7} -14&=&0\\2x_{2} x_{8} -24&=&0\\2x_{3} x_{9} -30&=&0\\2x_{3} x_{9} +2x_{5} x_{11} -34&=&0\\2x_{3} x_{9} +2x_{5} x_{11} +2x_{6} x_{12} -38&=&0\\2x_{4} x_{10} +2x_{5} x_{11} +2x_{6} x_{12} -40&=&0\\x_{4} x_{10} +2x_{5} x_{11} -20&=&0\\\end{array}

\begin{array}{rcl}h&=&20h_{7}+40h_{6}+38h_{5}+34h_{4}+30h_{3}+24h_{2}+14h_{1}\\e&=&x_{5} g_{23}+x_{4} g_{19}+x_{3} g_{16}+x_{6} g_{12}+x_{2} g_{2}+x_{1} g_{1}\\f&=&g_{-1}+g_{-2}+g_{-12}+g_{-16}+g_{-19}+g_{-23}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 2 & 0\\ 0 & 0 & 2 & 0 & 2 & 2\\ 0 & 0 & 0 & 2 & 2 & 2\\ 0 & 0 & 0 & 1 & 2 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}14\\ 24\\ 30\\ 34\\ 38\\ 40\\ 20\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&20h_{7}+40h_{6}+38h_{5}+34h_{4}+30h_{3}+24h_{2}+14h_{1}\\ e&=&x_{5} g_{23}+x_{4} g_{19}+x_{3} g_{16}+x_{6} g_{12}+x_{2} g_{2}+x_{1} g_{1}\\ f&=&x_{7} g_{-1}+x_{8} g_{-2}+x_{12} g_{-12}+x_{9} g_{-16}+x_{10} g_{-19}+x_{11} g_{-23}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{7} -14&=&0\\2x_{2} x_{8} -24&=&0\\2x_{3} x_{9} -30&=&0\\2x_{3} x_{9} +2x_{5} x_{11} -34&=&0\\2x_{3} x_{9} +2x_{5} x_{11} +2x_{6} x_{12} -38&=&0\\2x_{4} x_{10} +2x_{5} x_{11} +2x_{6} x_{12} -40&=&0\\x_{4} x_{10} +2x_{5} x_{11} -20&=&0\\\end{array}

A^{87}_1

h-characteristic: (2, 2, 2, 0, 0, 0, 2)
Length of the weight dual to h: 174
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 4
Containing regular semisimple subalgebra number 1:

C^{1}_4+3A^{1}_1

Containing regular semisimple subalgebra number 2:

C^{1}_4+A^{2}_1+A^{1}_1

Containing regular semisimple subalgebra number 3:

C^{1}_5+2A^{1}_1

Containing regular semisimple subalgebra number 4:

C^{1}_5+A^{2}_1

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

V_{14\psi}+V_{10\psi}+3V_{8\psi}+4V_{6\psi}+7V_{2\psi}+3V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&19h_{7}+36h_{6}+34h_{5}+32h_{4}+30h_{3}+24h_{2}+14h_{1}\\ e&=&g_{37}+g_{29}+15g_{22}+g_{19}+16g_{7}+12g_{2}+7g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-7}+g_{-19}+g_{-22}+g_{-29}+g_{-37}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&19h_{7}+36h_{6}+34h_{5}+32h_{4}+30h_{3}+24h_{2}+14h_{1}\\ [h, e]&=&2g_{37}+2g_{29}+30g_{22}+2g_{19}+32g_{7}+24g_{2}+14g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-7}-2g_{-19}-2g_{-22}-2g_{-29}-2g_{-37}\end{array}

Centralizer type:

A^{8}_1

Killing form square of Cartan element dual to ambient long root: 32
Basis of the centralizer (dimension: 3):

g_{4}-g_{-4}

g_{5}-g_{-5}

g_{11}-g_{-11}

Basis of centralizer intersected with cartan (dimension: 0):
Cartan of centralizer (dimension: 1):

g_{11}+g_{5}+g_{4}-g_{-4}-g_{-5}-g_{-11}

Cartan-generating semisimple element:

g_{11}+g_{5}+g_{4}-g_{-4}-g_{-5}-g_{-11}

adjoint action:

\begin{pmatrix}0 & -1 & 1\\ 1 & 0 & -1\\ -1 & 1 & 0\\ \end{pmatrix}

Characteristic polynomial ad H:

x^3+3x

Factorization of characteristic polynomial of ad H: (x )(x^2+3)
Eigenvalues of ad H:

0

\sqrt{-3}

-\sqrt{-3}

3 eigenvectors of ad H:

(1,1,1)

(−12−3−12,12−3−12,1)

(12−3−12,−12−3−12,1)

Centralizer type: A^{8}_1
Reductive components (1 total):
Scalar product computed:

\begin{pmatrix}1/64\\ \end{pmatrix}

Simple basis of Cartan of centralizer (1 total):

-2/3\sqrt{-3}g_{11}-2/3\sqrt{-3}g_{5}-2/3\sqrt{-3}g_{4}+2/3\sqrt{-3}g_{-4}+2/3\sqrt{-3}g_{-5}+2/3\sqrt{-3}g_{-11}

matching e:

g_{11}+\left(1/2\sqrt{-3}-1/2\right)g_{5}+\left(-1/2\sqrt{-3}-1/2\right)g_{4}+\left(1/2\sqrt{-3}+1/2\right)g_{-4}+\left(-1/2\sqrt{-3}+1/2\right)g_{-5}-g_{-11}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 2/3\sqrt{-3} & -2/3\sqrt{-3}\\ -2/3\sqrt{-3} & 0 & 2/3\sqrt{-3}\\ 2/3\sqrt{-3} & -2/3\sqrt{-3} & 0\\ \end{pmatrix}

Linear space basis of intersection of centralizer and ambient Cartan:

-2/3\sqrt{-3}g_{11}-2/3\sqrt{-3}g_{5}-2/3\sqrt{-3}g_{4}+2/3\sqrt{-3}g_{-4}+2/3\sqrt{-3}g_{-5}+2/3\sqrt{-3}g_{-11}

matching e:

g_{11}+\left(1/2\sqrt{-3}-1/2\right)g_{5}+\left(-1/2\sqrt{-3}-1/2\right)g_{4}+\left(1/2\sqrt{-3}+1/2\right)g_{-4}+\left(-1/2\sqrt{-3}+1/2\right)g_{-5}-g_{-11}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 2/3\sqrt{-3} & -2/3\sqrt{-3}\\ -2/3\sqrt{-3} & 0 & 2/3\sqrt{-3}\\ 2/3\sqrt{-3} & -2/3\sqrt{-3} & 0\\ \end{pmatrix}

Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form:

\begin{pmatrix}256\\ \end{pmatrix}

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&19h_{7}+36h_{6}+34h_{5}+32h_{4}+30h_{3}+24h_{2}+14h_{1}\\ e&=&x_{5} g_{37}+x_{6} g_{29}+x_{3} g_{22}+x_{7} g_{19}+x_{4} g_{7}+x_{2} g_{2}+x_{1} g_{1}\\ f&=&x_{8} g_{-1}+x_{9} g_{-2}+x_{11} g_{-7}+x_{14} g_{-19}+x_{10} g_{-22}+x_{13} g_{-29}+x_{12} g_{-37}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -19\right)h_{7}+\left(2x_{3} x_{10} +2x_{5} x_{12} +2x_{6} x_{13} +2x_{7} x_{14} -36\right)h_{6}+\left(2x_{3} x_{10} +2x_{5} x_{12} +2x_{6} x_{13} -34\right)h_{5}+\left(2x_{3} x_{10} +2x_{5} x_{12} -32\right)h_{4}+\left(2x_{3} x_{10} -30\right)h_{3}+\left(2x_{2} x_{9} -24\right)h_{2}+\left(2x_{1} x_{8} -14\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{8} -14&=&0\\2x_{2} x_{9} -24&=&0\\2x_{3} x_{10} -30&=&0\\2x_{3} x_{10} +2x_{5} x_{12} -32&=&0\\2x_{3} x_{10} +2x_{5} x_{12} +2x_{6} x_{13} -34&=&0\\2x_{3} x_{10} +2x_{5} x_{12} +2x_{6} x_{13} +2x_{7} x_{14} -36&=&0\\x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -19&=&0\\\end{array}

\begin{array}{rcl}h&=&19h_{7}+36h_{6}+34h_{5}+32h_{4}+30h_{3}+24h_{2}+14h_{1}\\e&=&x_{5} g_{37}+x_{6} g_{29}+x_{3} g_{22}+x_{7} g_{19}+x_{4} g_{7}+x_{2} g_{2}+x_{1} g_{1}\\f&=&g_{-1}+g_{-2}+g_{-7}+g_{-19}+g_{-22}+g_{-29}+g_{-37}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 2 & 0 & 0\\ 0 & 0 & 2 & 0 & 2 & 2 & 0\\ 0 & 0 & 2 & 0 & 2 & 2 & 2\\ 0 & 0 & 0 & 1 & 1 & 1 & 1\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}14\\ 24\\ 30\\ 32\\ 34\\ 36\\ 19\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&19h_{7}+36h_{6}+34h_{5}+32h_{4}+30h_{3}+24h_{2}+14h_{1}\\ e&=&x_{5} g_{37}+x_{6} g_{29}+x_{3} g_{22}+x_{7} g_{19}+x_{4} g_{7}+x_{2} g_{2}+x_{1} g_{1}\\ f&=&x_{8} g_{-1}+x_{9} g_{-2}+x_{11} g_{-7}+x_{14} g_{-19}+x_{10} g_{-22}+x_{13} g_{-29}+x_{12} g_{-37}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{8} -14&=&0\\2x_{2} x_{9} -24&=&0\\2x_{3} x_{10} -30&=&0\\2x_{3} x_{10} +2x_{5} x_{12} -32&=&0\\2x_{3} x_{10} +2x_{5} x_{12} +2x_{6} x_{13} -34&=&0\\2x_{3} x_{10} +2x_{5} x_{12} +2x_{6} x_{13} +2x_{7} x_{14} -36&=&0\\x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -19&=&0\\\end{array}

A^{86}_1

h-characteristic: (2, 2, 2, 0, 0, 1, 0)
Length of the weight dual to h: 172
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1:

C^{1}_4+2A^{1}_1

Containing regular semisimple subalgebra number 2:

C^{1}_4+A^{2}_1

Containing regular semisimple subalgebra number 3:

C^{1}_5+A^{1}_1

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

V_{14\psi}+V_{10\psi}+2V_{8\psi}+2V_{7\psi}+3V_{6\psi}+4V_{2\psi}+4V_{\psi}+4V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&18h_{7}+36h_{6}+34h_{5}+32h_{4}+30h_{3}+24h_{2}+14h_{1}\\ e&=&g_{37}+g_{29}+16g_{19}+15g_{16}+12g_{2}+7g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-16}+g_{-19}+g_{-29}+g_{-37}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&18h_{7}+36h_{6}+34h_{5}+32h_{4}+30h_{3}+24h_{2}+14h_{1}\\ [h, e]&=&2g_{37}+2g_{29}+32g_{19}+30g_{16}+24g_{2}+14g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-16}-2g_{-19}-2g_{-29}-2g_{-37}\end{array}

Centralizer type:

A_1

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&18h_{7}+36h_{6}+34h_{5}+32h_{4}+30h_{3}+24h_{2}+14h_{1}\\ e&=&x_{5} g_{37}+x_{6} g_{29}+x_{4} g_{19}+x_{3} g_{16}+x_{2} g_{2}+x_{1} g_{1}\\ f&=&x_{7} g_{-1}+x_{8} g_{-2}+x_{9} g_{-16}+x_{10} g_{-19}+x_{12} g_{-29}+x_{11} g_{-37}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -18\right)h_{7}+\left(2x_{4} x_{10} +2x_{5} x_{11} +2x_{6} x_{12} -36\right)h_{6}+\left(2x_{3} x_{9} +2x_{5} x_{11} +2x_{6} x_{12} -34\right)h_{5}+\left(2x_{3} x_{9} +2x_{5} x_{11} -32\right)h_{4}+\left(2x_{3} x_{9} -30\right)h_{3}+\left(2x_{2} x_{8} -24\right)h_{2}+\left(2x_{1} x_{7} -14\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{7} -14&=&0\\2x_{2} x_{8} -24&=&0\\2x_{3} x_{9} -30&=&0\\2x_{3} x_{9} +2x_{5} x_{11} -32&=&0\\2x_{3} x_{9} +2x_{5} x_{11} +2x_{6} x_{12} -34&=&0\\2x_{4} x_{10} +2x_{5} x_{11} +2x_{6} x_{12} -36&=&0\\x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -18&=&0\\\end{array}

\begin{array}{rcl}h&=&18h_{7}+36h_{6}+34h_{5}+32h_{4}+30h_{3}+24h_{2}+14h_{1}\\e&=&x_{5} g_{37}+x_{6} g_{29}+x_{4} g_{19}+x_{3} g_{16}+x_{2} g_{2}+x_{1} g_{1}\\f&=&g_{-1}+g_{-2}+g_{-16}+g_{-19}+g_{-29}+g_{-37}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 2 & 0\\ 0 & 0 & 2 & 0 & 2 & 2\\ 0 & 0 & 0 & 2 & 2 & 2\\ 0 & 0 & 0 & 1 & 1 & 1\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}14\\ 24\\ 30\\ 32\\ 34\\ 36\\ 18\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&18h_{7}+36h_{6}+34h_{5}+32h_{4}+30h_{3}+24h_{2}+14h_{1}\\ e&=&x_{5} g_{37}+x_{6} g_{29}+x_{4} g_{19}+x_{3} g_{16}+x_{2} g_{2}+x_{1} g_{1}\\ f&=&x_{7} g_{-1}+x_{8} g_{-2}+x_{9} g_{-16}+x_{10} g_{-19}+x_{12} g_{-29}+x_{11} g_{-37}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{7} -14&=&0\\2x_{2} x_{8} -24&=&0\\2x_{3} x_{9} -30&=&0\\2x_{3} x_{9} +2x_{5} x_{11} -32&=&0\\2x_{3} x_{9} +2x_{5} x_{11} +2x_{6} x_{12} -34&=&0\\2x_{4} x_{10} +2x_{5} x_{11} +2x_{6} x_{12} -36&=&0\\x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -18&=&0\\\end{array}

A^{85}_1

h-characteristic: (2, 2, 2, 0, 1, 0, 0)
Length of the weight dual to h: 170
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1:

C^{1}_4+A^{1}_1

Containing regular semisimple subalgebra number 2:

C^{1}_5

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

V_{14\psi}+V_{10\psi}+V_{8\psi}+4V_{7\psi}+2V_{6\psi}+2V_{2\psi}+4V_{\psi}+10V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&17h_{7}+34h_{6}+34h_{5}+32h_{4}+30h_{3}+24h_{2}+14h_{1}\\ e&=&g_{37}+16g_{29}+15g_{10}+12g_{2}+7g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-10}+g_{-29}+g_{-37}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&17h_{7}+34h_{6}+34h_{5}+32h_{4}+30h_{3}+24h_{2}+14h_{1}\\ [h, e]&=&2g_{37}+32g_{29}+30g_{10}+24g_{2}+14g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-10}-2g_{-29}-2g_{-37}\end{array}

Centralizer type:

B_2

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&17h_{7}+34h_{6}+34h_{5}+32h_{4}+30h_{3}+24h_{2}+14h_{1}\\ e&=&x_{5} g_{37}+x_{4} g_{29}+x_{3} g_{10}+x_{2} g_{2}+x_{1} g_{1}\\ f&=&x_{6} g_{-1}+x_{7} g_{-2}+x_{8} g_{-10}+x_{9} g_{-29}+x_{10} g_{-37}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(x_{4} x_{9} +x_{5} x_{10} -17\right)h_{7}+\left(2x_{4} x_{9} +2x_{5} x_{10} -34\right)h_{6}+\left(2x_{4} x_{9} +2x_{5} x_{10} -34\right)h_{5}+\left(2x_{3} x_{8} +2x_{5} x_{10} -32\right)h_{4}+\left(2x_{3} x_{8} -30\right)h_{3}+\left(2x_{2} x_{7} -24\right)h_{2}+\left(2x_{1} x_{6} -14\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{6} -14&=&0\\2x_{2} x_{7} -24&=&0\\2x_{3} x_{8} -30&=&0\\2x_{3} x_{8} +2x_{5} x_{10} -32&=&0\\2x_{4} x_{9} +2x_{5} x_{10} -34&=&0\\2x_{4} x_{9} +2x_{5} x_{10} -34&=&0\\x_{4} x_{9} +x_{5} x_{10} -17&=&0\\\end{array}

\begin{array}{rcl}h&=&17h_{7}+34h_{6}+34h_{5}+32h_{4}+30h_{3}+24h_{2}+14h_{1}\\e&=&x_{5} g_{37}+x_{4} g_{29}+x_{3} g_{10}+x_{2} g_{2}+x_{1} g_{1}\\f&=&g_{-1}+g_{-2}+g_{-10}+g_{-29}+g_{-37}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 2 & 0 & 2\\ 0 & 0 & 0 & 2 & 2\\ 0 & 0 & 0 & 2 & 2\\ 0 & 0 & 0 & 1 & 1\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}14\\ 24\\ 30\\ 32\\ 34\\ 34\\ 17\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&17h_{7}+34h_{6}+34h_{5}+32h_{4}+30h_{3}+24h_{2}+14h_{1}\\ e&=&x_{5} g_{37}+x_{4} g_{29}+x_{3} g_{10}+x_{2} g_{2}+x_{1} g_{1}\\ f&=&x_{6} g_{-1}+x_{7} g_{-2}+x_{8} g_{-10}+x_{9} g_{-29}+x_{10} g_{-37}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{6} -14&=&0\\2x_{2} x_{7} -24&=&0\\2x_{3} x_{8} -30&=&0\\2x_{3} x_{8} +2x_{5} x_{10} -32&=&0\\2x_{4} x_{9} +2x_{5} x_{10} -34&=&0\\2x_{4} x_{9} +2x_{5} x_{10} -34&=&0\\x_{4} x_{9} +x_{5} x_{10} -17&=&0\\\end{array}

A^{84}_1

h-characteristic: (2, 2, 2, 1, 0, 0, 0)
Length of the weight dual to h: 168
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1:

C^{1}_4

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

V_{14\psi}+V_{10\psi}+6V_{7\psi}+V_{6\psi}+V_{2\psi}+21V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&16h_{7}+32h_{6}+32h_{5}+32h_{4}+30h_{3}+24h_{2}+14h_{1}\\ e&=&16g_{37}+15g_{3}+12g_{2}+7g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-3}+g_{-37}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&16h_{7}+32h_{6}+32h_{5}+32h_{4}+30h_{3}+24h_{2}+14h_{1}\\ [h, e]&=&32g_{37}+30g_{3}+24g_{2}+14g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-3}-2g_{-37}\end{array}

Centralizer type:

C_3

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&16h_{7}+32h_{6}+32h_{5}+32h_{4}+30h_{3}+24h_{2}+14h_{1}\\ e&=&x_{4} g_{37}+x_{3} g_{3}+x_{2} g_{2}+x_{1} g_{1}\\ f&=&x_{5} g_{-1}+x_{6} g_{-2}+x_{7} g_{-3}+x_{8} g_{-37}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(x_{4} x_{8} -16\right)h_{7}+\left(2x_{4} x_{8} -32\right)h_{6}+\left(2x_{4} x_{8} -32\right)h_{5}+\left(2x_{4} x_{8} -32\right)h_{4}+\left(2x_{3} x_{7} -30\right)h_{3}+\left(2x_{2} x_{6} -24\right)h_{2}+\left(2x_{1} x_{5} -14\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{5} -14&=&0\\2x_{2} x_{6} -24&=&0\\2x_{3} x_{7} -30&=&0\\2x_{4} x_{8} -32&=&0\\2x_{4} x_{8} -32&=&0\\2x_{4} x_{8} -32&=&0\\x_{4} x_{8} -16&=&0\\\end{array}

\begin{array}{rcl}h&=&16h_{7}+32h_{6}+32h_{5}+32h_{4}+30h_{3}+24h_{2}+14h_{1}\\e&=&x_{4} g_{37}+x_{3} g_{3}+x_{2} g_{2}+x_{1} g_{1}\\f&=&g_{-1}+g_{-2}+g_{-3}+g_{-37}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0 & 0\\ 0 & 2 & 0 & 0\\ 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 2\\ 0 & 0 & 0 & 2\\ 0 & 0 & 0 & 2\\ 0 & 0 & 0 & 1\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}14\\ 24\\ 30\\ 32\\ 32\\ 32\\ 16\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&16h_{7}+32h_{6}+32h_{5}+32h_{4}+30h_{3}+24h_{2}+14h_{1}\\ e&=&x_{4} g_{37}+x_{3} g_{3}+x_{2} g_{2}+x_{1} g_{1}\\ f&=&x_{5} g_{-1}+x_{6} g_{-2}+x_{7} g_{-3}+x_{8} g_{-37}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{5} -14&=&0\\2x_{2} x_{6} -24&=&0\\2x_{3} x_{7} -30&=&0\\2x_{4} x_{8} -32&=&0\\2x_{4} x_{8} -32&=&0\\2x_{4} x_{8} -32&=&0\\x_{4} x_{8} -16&=&0\\\end{array}

A^{71}_1

h-characteristic: (0, 2, 0, 2, 0, 0, 2)
Length of the weight dual to h: 142
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1:

2C^{1}_3+A^{1}_1

Containing regular semisimple subalgebra number 2:

A^{2}_5+A^{1}_1

Containing regular semisimple subalgebra number 3:

C^{1}_4+C^{1}_3

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

3V_{10\psi}+V_{8\psi}+5V_{6\psi}+3V_{4\psi}+4V_{2\psi}+V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&19h_{7}+36h_{6}+34h_{5}+32h_{4}+26h_{3}+20h_{2}+10h_{1}\\ e&=&g_{29}+9g_{19}+8g_{17}+8g_{16}+5g_{14}+9g_{7}+5g_{2}\\ f&=&g_{-2}+g_{-7}+g_{-14}+g_{-16}+g_{-17}+g_{-19}+g_{-29}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&19h_{7}+36h_{6}+34h_{5}+32h_{4}+26h_{3}+20h_{2}+10h_{1}\\ [h, e]&=&2g_{29}+18g_{19}+16g_{17}+16g_{16}+10g_{14}+18g_{7}+10g_{2}\\ [h, f]&=&-2g_{-2}-2g_{-7}-2g_{-14}-2g_{-16}-2g_{-17}-2g_{-19}-2g_{-29}\end{array}

Centralizer type:

0

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&19h_{7}+36h_{6}+34h_{5}+32h_{4}+26h_{3}+20h_{2}+10h_{1}\\ e&=&x_{7} g_{29}+x_{6} g_{19}+x_{2} g_{17}+x_{5} g_{16}+x_{1} g_{14}+x_{3} g_{7}+x_{4} g_{2}\\ f&=&x_{11} g_{-2}+x_{10} g_{-7}+x_{8} g_{-14}+x_{12} g_{-16}+x_{9} g_{-17}+x_{13} g_{-19}+x_{14} g_{-29}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(x_{3} x_{10} +x_{6} x_{13} +x_{7} x_{14} -19\right)h_{7}+\left(2x_{2} x_{9} +2x_{6} x_{13} +2x_{7} x_{14} -36\right)h_{6}+\left(2x_{2} x_{9} +2x_{5} x_{12} +2x_{7} x_{14} -34\right)h_{5}+\left(2x_{2} x_{9} +2x_{5} x_{12} -32\right)h_{4}+\left(2x_{1} x_{8} +2x_{5} x_{12} -26\right)h_{3}+\left(2x_{1} x_{8} +2x_{4} x_{11} -20\right)h_{2}+\left(2x_{1} x_{8} -10\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{8} -10&=&0\\2x_{1} x_{8} +2x_{4} x_{11} -20&=&0\\2x_{1} x_{8} +2x_{5} x_{12} -26&=&0\\2x_{2} x_{9} +2x_{5} x_{12} -32&=&0\\2x_{2} x_{9} +2x_{5} x_{12} +2x_{7} x_{14} -34&=&0\\2x_{2} x_{9} +2x_{6} x_{13} +2x_{7} x_{14} -36&=&0\\x_{3} x_{10} +x_{6} x_{13} +x_{7} x_{14} -19&=&0\\\end{array}

\begin{array}{rcl}h&=&19h_{7}+36h_{6}+34h_{5}+32h_{4}+26h_{3}+20h_{2}+10h_{1}\\e&=&x_{7} g_{29}+x_{6} g_{19}+x_{2} g_{17}+x_{5} g_{16}+x_{1} g_{14}+x_{3} g_{7}+x_{4} g_{2}\\f&=&g_{-2}+g_{-7}+g_{-14}+g_{-16}+g_{-17}+g_{-19}+g_{-29}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0\\ 2 & 0 & 0 & 2 & 0 & 0 & 0\\ 2 & 0 & 0 & 0 & 2 & 0 & 0\\ 0 & 2 & 0 & 0 & 2 & 0 & 0\\ 0 & 2 & 0 & 0 & 2 & 0 & 2\\ 0 & 2 & 0 & 0 & 0 & 2 & 2\\ 0 & 0 & 1 & 0 & 0 & 1 & 1\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}10\\ 20\\ 26\\ 32\\ 34\\ 36\\ 19\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&19h_{7}+36h_{6}+34h_{5}+32h_{4}+26h_{3}+20h_{2}+10h_{1}\\ e&=&x_{7} g_{29}+x_{6} g_{19}+x_{2} g_{17}+x_{5} g_{16}+x_{1} g_{14}+x_{3} g_{7}+x_{4} g_{2}\\ f&=&x_{11} g_{-2}+x_{10} g_{-7}+x_{8} g_{-14}+x_{12} g_{-16}+x_{9} g_{-17}+x_{13} g_{-19}+x_{14} g_{-29}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{8} -10&=&0\\2x_{1} x_{8} +2x_{4} x_{11} -20&=&0\\2x_{1} x_{8} +2x_{5} x_{12} -26&=&0\\2x_{2} x_{9} +2x_{5} x_{12} -32&=&0\\2x_{2} x_{9} +2x_{5} x_{12} +2x_{7} x_{14} -34&=&0\\2x_{2} x_{9} +2x_{6} x_{13} +2x_{7} x_{14} -36&=&0\\x_{3} x_{10} +x_{6} x_{13} +x_{7} x_{14} -19&=&0\\\end{array}

A^{70}_1

h-characteristic: (0, 2, 0, 2, 0, 1, 0)
Length of the weight dual to h: 140
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1:

2C^{1}_3

Containing regular semisimple subalgebra number 2:

A^{2}_5

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

3V_{10\psi}+V_{8\psi}+3V_{6\psi}+4V_{5\psi}+V_{4\psi}+3V_{2\psi}+4V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&18h_{7}+36h_{6}+34h_{5}+32h_{4}+26h_{3}+20h_{2}+10h_{1}\\ e&=&9g_{29}+9g_{19}+5g_{14}+8g_{11}+8g_{10}+5g_{2}\\ f&=&g_{-2}+g_{-10}+g_{-11}+g_{-14}+g_{-19}+g_{-29}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&18h_{7}+36h_{6}+34h_{5}+32h_{4}+26h_{3}+20h_{2}+10h_{1}\\ [h, e]&=&18g_{29}+18g_{19}+10g_{14}+16g_{11}+16g_{10}+10g_{2}\\ [h, f]&=&-2g_{-2}-2g_{-10}-2g_{-11}-2g_{-14}-2g_{-19}-2g_{-29}\end{array}

Centralizer type:

A_1

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&18h_{7}+36h_{6}+34h_{5}+32h_{4}+26h_{3}+20h_{2}+10h_{1}\\ e&=&x_{6} g_{29}+x_{3} g_{19}+x_{1} g_{14}+x_{2} g_{11}+x_{5} g_{10}+x_{4} g_{2}\\ f&=&x_{10} g_{-2}+x_{11} g_{-10}+x_{8} g_{-11}+x_{7} g_{-14}+x_{9} g_{-19}+x_{12} g_{-29}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(x_{3} x_{9} +x_{6} x_{12} -18\right)h_{7}+\left(2x_{3} x_{9} +2x_{6} x_{12} -36\right)h_{6}+\left(2x_{2} x_{8} +2x_{6} x_{12} -34\right)h_{5}+\left(2x_{2} x_{8} +2x_{5} x_{11} -32\right)h_{4}+\left(2x_{1} x_{7} +2x_{5} x_{11} -26\right)h_{3}+\left(2x_{1} x_{7} +2x_{4} x_{10} -20\right)h_{2}+\left(2x_{1} x_{7} -10\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{7} -10&=&0\\2x_{1} x_{7} +2x_{4} x_{10} -20&=&0\\2x_{1} x_{7} +2x_{5} x_{11} -26&=&0\\2x_{2} x_{8} +2x_{5} x_{11} -32&=&0\\2x_{2} x_{8} +2x_{6} x_{12} -34&=&0\\2x_{3} x_{9} +2x_{6} x_{12} -36&=&0\\x_{3} x_{9} +x_{6} x_{12} -18&=&0\\\end{array}

\begin{array}{rcl}h&=&18h_{7}+36h_{6}+34h_{5}+32h_{4}+26h_{3}+20h_{2}+10h_{1}\\e&=&x_{6} g_{29}+x_{3} g_{19}+x_{1} g_{14}+x_{2} g_{11}+x_{5} g_{10}+x_{4} g_{2}\\f&=&g_{-2}+g_{-10}+g_{-11}+g_{-14}+g_{-19}+g_{-29}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0\\ 2 & 0 & 0 & 2 & 0 & 0\\ 2 & 0 & 0 & 0 & 2 & 0\\ 0 & 2 & 0 & 0 & 2 & 0\\ 0 & 2 & 0 & 0 & 0 & 2\\ 0 & 0 & 2 & 0 & 0 & 2\\ 0 & 0 & 1 & 0 & 0 & 1\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}10\\ 20\\ 26\\ 32\\ 34\\ 36\\ 18\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&18h_{7}+36h_{6}+34h_{5}+32h_{4}+26h_{3}+20h_{2}+10h_{1}\\ e&=&x_{6} g_{29}+x_{3} g_{19}+x_{1} g_{14}+x_{2} g_{11}+x_{5} g_{10}+x_{4} g_{2}\\ f&=&x_{10} g_{-2}+x_{11} g_{-10}+x_{8} g_{-11}+x_{7} g_{-14}+x_{9} g_{-19}+x_{12} g_{-29}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{7} -10&=&0\\2x_{1} x_{7} +2x_{4} x_{10} -20&=&0\\2x_{1} x_{7} +2x_{5} x_{11} -26&=&0\\2x_{2} x_{8} +2x_{5} x_{11} -32&=&0\\2x_{2} x_{8} +2x_{6} x_{12} -34&=&0\\2x_{3} x_{9} +2x_{6} x_{12} -36&=&0\\x_{3} x_{9} +x_{6} x_{12} -18&=&0\\\end{array}

A^{55}_1

h-characteristic: (2, 0, 0, 2, 0, 0, 2)
Length of the weight dual to h: 110
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1:

C^{1}_3+2B^{1}_2

Containing regular semisimple subalgebra number 2:

A^{2}_3+C^{1}_3

Containing regular semisimple subalgebra number 3:

C^{1}_5+B^{1}_2

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

V_{10\psi}+2V_{8\psi}+6V_{6\psi}+3V_{4\psi}+6V_{2\psi}+V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&17h_{7}+32h_{6}+30h_{5}+28h_{4}+22h_{3}+16h_{2}+10h_{1}\\ e&=&4g_{29}+4g_{19}+8g_{17}+3g_{16}+3g_{15}+5g_{14}+9g_{7}\\ f&=&g_{-7}+g_{-14}+g_{-15}+g_{-16}+g_{-17}+g_{-19}+g_{-29}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&17h_{7}+32h_{6}+30h_{5}+28h_{4}+22h_{3}+16h_{2}+10h_{1}\\ [h, e]&=&8g_{29}+8g_{19}+16g_{17}+6g_{16}+6g_{15}+10g_{14}+18g_{7}\\ [h, f]&=&-2g_{-7}-2g_{-14}-2g_{-15}-2g_{-16}-2g_{-17}-2g_{-19}-2g_{-29}\end{array}

Centralizer type:

0

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&17h_{7}+32h_{6}+30h_{5}+28h_{4}+22h_{3}+16h_{2}+10h_{1}\\ e&=&x_{4} g_{29}+x_{6} g_{19}+x_{2} g_{17}+x_{7} g_{16}+x_{5} g_{15}+x_{1} g_{14}+x_{3} g_{7}\\ f&=&x_{10} g_{-7}+x_{8} g_{-14}+x_{12} g_{-15}+x_{14} g_{-16}+x_{9} g_{-17}+x_{13} g_{-19}+x_{11} g_{-29}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(x_{3} x_{10} +x_{4} x_{11} +x_{6} x_{13} -17\right)h_{7}+\left(2x_{2} x_{9} +2x_{4} x_{11} +2x_{6} x_{13} -32\right)h_{6}+\left(2x_{2} x_{9} +2x_{4} x_{11} +2x_{7} x_{14} -30\right)h_{5}+\left(2x_{2} x_{9} +2x_{5} x_{12} +2x_{7} x_{14} -28\right)h_{4}+\left(2x_{1} x_{8} +2x_{5} x_{12} +2x_{7} x_{14} -22\right)h_{3}+\left(2x_{1} x_{8} +2x_{5} x_{12} -16\right)h_{2}+\left(2x_{1} x_{8} -10\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{8} -10&=&0\\2x_{1} x_{8} +2x_{5} x_{12} -16&=&0\\2x_{1} x_{8} +2x_{5} x_{12} +2x_{7} x_{14} -22&=&0\\2x_{2} x_{9} +2x_{5} x_{12} +2x_{7} x_{14} -28&=&0\\2x_{2} x_{9} +2x_{4} x_{11} +2x_{7} x_{14} -30&=&0\\2x_{2} x_{9} +2x_{4} x_{11} +2x_{6} x_{13} -32&=&0\\x_{3} x_{10} +x_{4} x_{11} +x_{6} x_{13} -17&=&0\\\end{array}

\begin{array}{rcl}h&=&17h_{7}+32h_{6}+30h_{5}+28h_{4}+22h_{3}+16h_{2}+10h_{1}\\e&=&x_{4} g_{29}+x_{6} g_{19}+x_{2} g_{17}+x_{7} g_{16}+x_{5} g_{15}+x_{1} g_{14}+x_{3} g_{7}\\f&=&g_{-7}+g_{-14}+g_{-15}+g_{-16}+g_{-17}+g_{-19}+g_{-29}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0\\ 2 & 0 & 0 & 0 & 2 & 0 & 0\\ 2 & 0 & 0 & 0 & 2 & 0 & 2\\ 0 & 2 & 0 & 0 & 2 & 0 & 2\\ 0 & 2 & 0 & 2 & 0 & 0 & 2\\ 0 & 2 & 0 & 2 & 0 & 2 & 0\\ 0 & 0 & 1 & 1 & 0 & 1 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}10\\ 16\\ 22\\ 28\\ 30\\ 32\\ 17\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&17h_{7}+32h_{6}+30h_{5}+28h_{4}+22h_{3}+16h_{2}+10h_{1}\\ e&=&x_{4} g_{29}+x_{6} g_{19}+x_{2} g_{17}+x_{7} g_{16}+x_{5} g_{15}+x_{1} g_{14}+x_{3} g_{7}\\ f&=&x_{10} g_{-7}+x_{8} g_{-14}+x_{12} g_{-15}+x_{14} g_{-16}+x_{9} g_{-17}+x_{13} g_{-19}+x_{11} g_{-29}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{8} -10&=&0\\2x_{1} x_{8} +2x_{5} x_{12} -16&=&0\\2x_{1} x_{8} +2x_{5} x_{12} +2x_{7} x_{14} -22&=&0\\2x_{2} x_{9} +2x_{5} x_{12} +2x_{7} x_{14} -28&=&0\\2x_{2} x_{9} +2x_{4} x_{11} +2x_{7} x_{14} -30&=&0\\2x_{2} x_{9} +2x_{4} x_{11} +2x_{6} x_{13} -32&=&0\\x_{3} x_{10} +x_{4} x_{11} +x_{6} x_{13} -17&=&0\\\end{array}

A^{50}_1

h-characteristic: (0, 1, 1, 0, 1, 1, 0)
Length of the weight dual to h: 100
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1:

A^{2}_4+B^{1}_2

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

3V_{8\psi}+2V_{7\psi}+2V_{6\psi}+2V_{5\psi}+3V_{4\psi}+2V_{3\psi}+2V_{2\psi}+2V_{\psi}+3V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&16h_{7}+32h_{6}+30h_{5}+26h_{4}+22h_{3}+16h_{2}+8h_{1}\\ e&=&4g_{20}+4g_{19}+6g_{18}+6g_{17}+3g_{16}+4g_{9}\\ f&=&g_{-9}+g_{-16}+g_{-17}+g_{-18}+g_{-19}+g_{-20}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&16h_{7}+32h_{6}+30h_{5}+26h_{4}+22h_{3}+16h_{2}+8h_{1}\\ [h, e]&=&8g_{20}+8g_{19}+12g_{18}+12g_{17}+6g_{16}+8g_{9}\\ [h, f]&=&-2g_{-9}-2g_{-16}-2g_{-17}-2g_{-18}-2g_{-19}-2g_{-20}\end{array}

Centralizer type:

A^{5}_1

Killing form square of Cartan element dual to ambient long root: 32
Basis of the centralizer (dimension: 3):

h_{7}+2h_{4}-2h_{1}

g_{1}+g_{-4}+g_{-7}

g_{7}+g_{4}+g_{-1}

Basis of centralizer intersected with cartan (dimension: 1):

1/2h_{7}+h_{4}-h_{1}

Cartan of centralizer (dimension: 1):

1/2h_{7}+h_{4}-h_{1}

Cartan-generating semisimple element:

1/2h_{7}+h_{4}-h_{1}

adjoint action:

\begin{pmatrix}0 & 0 & 0\\ 0 & -1 & 0\\ 0 & 0 & 1\\ \end{pmatrix}

Characteristic polynomial ad H:

x^3-x

Factorization of characteristic polynomial of ad H: (x )(x -1)(x +1)
Eigenvalues of ad H:

0

1

-1

3 eigenvectors of ad H:

(1,0,0)

(0,0,1)

(0,1,0)

Centralizer type: A^{5}_1
Reductive components (1 total):
Scalar product computed:

\begin{pmatrix}1/40\\ \end{pmatrix}

Simple basis of Cartan of centralizer (1 total):

h_{7}+2h_{4}-2h_{1}

matching e:

g_{7}+g_{4}+g_{-1}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0\\ 0 & -2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}

Linear space basis of intersection of centralizer and ambient Cartan:

h_{7}+2h_{4}-2h_{1}

matching e:

g_{7}+g_{4}+g_{-1}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0\\ 0 & -2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}

Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form:

\begin{pmatrix}160\\ \end{pmatrix}

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&16h_{7}+32h_{6}+30h_{5}+26h_{4}+22h_{3}+16h_{2}+8h_{1}\\ e&=&x_{1} g_{20}+x_{5} g_{19}+x_{2} g_{18}+x_{3} g_{17}+x_{6} g_{16}+x_{4} g_{9}\\ f&=&x_{10} g_{-9}+x_{12} g_{-16}+x_{9} g_{-17}+x_{8} g_{-18}+x_{11} g_{-19}+x_{7} g_{-20}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(2x_{2} x_{8} +x_{5} x_{11} -16\right)h_{7}+\left(2x_{2} x_{8} +2x_{3} x_{9} +2x_{5} x_{11} -32\right)h_{6}+\left(2x_{2} x_{8} +2x_{3} x_{9} +2x_{6} x_{12} -30\right)h_{5}+\left(2x_{1} x_{7} +2x_{3} x_{9} +2x_{6} x_{12} -26\right)h_{4}+\left(2x_{1} x_{7} +2x_{4} x_{10} +2x_{6} x_{12} -22\right)h_{3}+\left(2x_{1} x_{7} +2x_{4} x_{10} -16\right)h_{2}+\left(2x_{1} x_{7} -8\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{7} -8&=&0\\2x_{1} x_{7} +2x_{4} x_{10} -16&=&0\\2x_{1} x_{7} +2x_{4} x_{10} +2x_{6} x_{12} -22&=&0\\2x_{1} x_{7} +2x_{3} x_{9} +2x_{6} x_{12} -26&=&0\\2x_{2} x_{8} +2x_{3} x_{9} +2x_{6} x_{12} -30&=&0\\2x_{2} x_{8} +2x_{3} x_{9} +2x_{5} x_{11} -32&=&0\\2x_{2} x_{8} +x_{5} x_{11} -16&=&0\\\end{array}

\begin{array}{rcl}h&=&16h_{7}+32h_{6}+30h_{5}+26h_{4}+22h_{3}+16h_{2}+8h_{1}\\e&=&x_{1} g_{20}+x_{5} g_{19}+x_{2} g_{18}+x_{3} g_{17}+x_{6} g_{16}+x_{4} g_{9}\\f&=&g_{-9}+g_{-16}+g_{-17}+g_{-18}+g_{-19}+g_{-20}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0\\ 2 & 0 & 0 & 2 & 0 & 0\\ 2 & 0 & 0 & 2 & 0 & 2\\ 2 & 0 & 2 & 0 & 0 & 2\\ 0 & 2 & 2 & 0 & 0 & 2\\ 0 & 2 & 2 & 0 & 2 & 0\\ 0 & 2 & 0 & 0 & 1 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}8\\ 16\\ 22\\ 26\\ 30\\ 32\\ 16\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&16h_{7}+32h_{6}+30h_{5}+26h_{4}+22h_{3}+16h_{2}+8h_{1}\\ e&=&x_{1} g_{20}+x_{5} g_{19}+x_{2} g_{18}+x_{3} g_{17}+x_{6} g_{16}+x_{4} g_{9}\\ f&=&x_{10} g_{-9}+x_{12} g_{-16}+x_{9} g_{-17}+x_{8} g_{-18}+x_{11} g_{-19}+x_{7} g_{-20}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{7} -8&=&0\\2x_{1} x_{7} +2x_{4} x_{10} -16&=&0\\2x_{1} x_{7} +2x_{4} x_{10} +2x_{6} x_{12} -22&=&0\\2x_{1} x_{7} +2x_{3} x_{9} +2x_{6} x_{12} -26&=&0\\2x_{2} x_{8} +2x_{3} x_{9} +2x_{6} x_{12} -30&=&0\\2x_{2} x_{8} +2x_{3} x_{9} +2x_{5} x_{11} -32&=&0\\2x_{2} x_{8} +x_{5} x_{11} -16&=&0\\\end{array}

A^{47}_1

h-characteristic: (2, 0, 2, 0, 0, 0, 2)
Length of the weight dual to h: 94
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 8
Containing regular semisimple subalgebra number 1:

C^{1}_3+B^{1}_2+2A^{1}_1

Containing regular semisimple subalgebra number 2:

C^{1}_3+B^{1}_2+A^{2}_1

Containing regular semisimple subalgebra number 3:

C^{1}_6+A^{1}_1

Containing regular semisimple subalgebra number 4:

C^{1}_5+2A^{1}_1

Containing regular semisimple subalgebra number 5:

C^{1}_4+C^{1}_3

Containing regular semisimple subalgebra number 6:

C^{1}_4+B^{1}_2+A^{1}_1

Containing regular semisimple subalgebra number 7:

2C^{1}_3+A^{1}_1

Containing regular semisimple subalgebra number 8:

C^{1}_5+A^{2}_1

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

V_{10\psi}+V_{8\psi}+5V_{6\psi}+5V_{4\psi}+8V_{2\psi}+V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&15h_{7}+28h_{6}+26h_{5}+24h_{4}+22h_{3}+16h_{2}+10h_{1}\\ e&=&4g_{37}+g_{29}+8g_{22}+g_{19}+3g_{9}+5g_{8}+9g_{7}\\ f&=&g_{-7}+g_{-8}+g_{-9}+g_{-19}+g_{-22}+g_{-29}+g_{-37}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&15h_{7}+28h_{6}+26h_{5}+24h_{4}+22h_{3}+16h_{2}+10h_{1}\\ [h, e]&=&8g_{37}+2g_{29}+16g_{22}+2g_{19}+6g_{9}+10g_{8}+18g_{7}\\ [h, f]&=&-2g_{-7}-2g_{-8}-2g_{-9}-2g_{-19}-2g_{-22}-2g_{-29}-2g_{-37}\end{array}

Centralizer type:

0

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&15h_{7}+28h_{6}+26h_{5}+24h_{4}+22h_{3}+16h_{2}+10h_{1}\\ e&=&x_{4} g_{37}+x_{6} g_{29}+x_{2} g_{22}+x_{7} g_{19}+x_{5} g_{9}+x_{1} g_{8}+x_{3} g_{7}\\ f&=&x_{10} g_{-7}+x_{8} g_{-8}+x_{12} g_{-9}+x_{14} g_{-19}+x_{9} g_{-22}+x_{13} g_{-29}+x_{11} g_{-37}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(x_{3} x_{10} +x_{4} x_{11} +x_{6} x_{13} +x_{7} x_{14} -15\right)h_{7}+\left(2x_{2} x_{9} +2x_{4} x_{11} +2x_{6} x_{13} +2x_{7} x_{14} -28\right)h_{6}+\left(2x_{2} x_{9} +2x_{4} x_{11} +2x_{6} x_{13} -26\right)h_{5}+\left(2x_{2} x_{9} +2x_{4} x_{11} -24\right)h_{4}+\left(2x_{2} x_{9} +2x_{5} x_{12} -22\right)h_{3}+\left(2x_{1} x_{8} +2x_{5} x_{12} -16\right)h_{2}+\left(2x_{1} x_{8} -10\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{8} -10&=&0\\2x_{1} x_{8} +2x_{5} x_{12} -16&=&0\\2x_{2} x_{9} +2x_{5} x_{12} -22&=&0\\2x_{2} x_{9} +2x_{4} x_{11} -24&=&0\\2x_{2} x_{9} +2x_{4} x_{11} +2x_{6} x_{13} -26&=&0\\2x_{2} x_{9} +2x_{4} x_{11} +2x_{6} x_{13} +2x_{7} x_{14} -28&=&0\\x_{3} x_{10} +x_{4} x_{11} +x_{6} x_{13} +x_{7} x_{14} -15&=&0\\\end{array}

\begin{array}{rcl}h&=&15h_{7}+28h_{6}+26h_{5}+24h_{4}+22h_{3}+16h_{2}+10h_{1}\\e&=&x_{4} g_{37}+x_{6} g_{29}+x_{2} g_{22}+x_{7} g_{19}+x_{5} g_{9}+x_{1} g_{8}+x_{3} g_{7}\\f&=&g_{-7}+g_{-8}+g_{-9}+g_{-19}+g_{-22}+g_{-29}+g_{-37}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0\\ 2 & 0 & 0 & 0 & 2 & 0 & 0\\ 0 & 2 & 0 & 0 & 2 & 0 & 0\\ 0 & 2 & 0 & 2 & 0 & 0 & 0\\ 0 & 2 & 0 & 2 & 0 & 2 & 0\\ 0 & 2 & 0 & 2 & 0 & 2 & 2\\ 0 & 0 & 1 & 1 & 0 & 1 & 1\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}10\\ 16\\ 22\\ 24\\ 26\\ 28\\ 15\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&15h_{7}+28h_{6}+26h_{5}+24h_{4}+22h_{3}+16h_{2}+10h_{1}\\ e&=&x_{4} g_{37}+x_{6} g_{29}+x_{2} g_{22}+x_{7} g_{19}+x_{5} g_{9}+x_{1} g_{8}+x_{3} g_{7}\\ f&=&x_{10} g_{-7}+x_{8} g_{-8}+x_{12} g_{-9}+x_{14} g_{-19}+x_{9} g_{-22}+x_{13} g_{-29}+x_{11} g_{-37}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{8} -10&=&0\\2x_{1} x_{8} +2x_{5} x_{12} -16&=&0\\2x_{2} x_{9} +2x_{5} x_{12} -22&=&0\\2x_{2} x_{9} +2x_{4} x_{11} -24&=&0\\2x_{2} x_{9} +2x_{4} x_{11} +2x_{6} x_{13} -26&=&0\\2x_{2} x_{9} +2x_{4} x_{11} +2x_{6} x_{13} +2x_{7} x_{14} -28&=&0\\x_{3} x_{10} +x_{4} x_{11} +x_{6} x_{13} +x_{7} x_{14} -15&=&0\\\end{array}

A^{46}_1

h-characteristic: (2, 0, 2, 0, 0, 1, 0)
Length of the weight dual to h: 92
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 5
Containing regular semisimple subalgebra number 1:

C^{1}_3+B^{1}_2+A^{1}_1

Containing regular semisimple subalgebra number 2:

C^{1}_6

Containing regular semisimple subalgebra number 3:

C^{1}_5+A^{1}_1

Containing regular semisimple subalgebra number 4:

C^{1}_4+B^{1}_2

Containing regular semisimple subalgebra number 5:

2C^{1}_3

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

V_{10\psi}+V_{8\psi}+4V_{6\psi}+2V_{5\psi}+3V_{4\psi}+2V_{3\psi}+5V_{2\psi}+2V_{\psi}+3V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&14h_{7}+28h_{6}+26h_{5}+24h_{4}+22h_{3}+16h_{2}+10h_{1}\\ e&=&4g_{37}+g_{29}+9g_{19}+8g_{16}+3g_{9}+5g_{8}\\ f&=&g_{-8}+g_{-9}+g_{-16}+g_{-19}+g_{-29}+g_{-37}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&14h_{7}+28h_{6}+26h_{5}+24h_{4}+22h_{3}+16h_{2}+10h_{1}\\ [h, e]&=&8g_{37}+2g_{29}+18g_{19}+16g_{16}+6g_{9}+10g_{8}\\ [h, f]&=&-2g_{-8}-2g_{-9}-2g_{-16}-2g_{-19}-2g_{-29}-2g_{-37}\end{array}

Centralizer type:

A_1

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&14h_{7}+28h_{6}+26h_{5}+24h_{4}+22h_{3}+16h_{2}+10h_{1}\\ e&=&x_{4} g_{37}+x_{6} g_{29}+x_{3} g_{19}+x_{2} g_{16}+x_{5} g_{9}+x_{1} g_{8}\\ f&=&x_{7} g_{-8}+x_{11} g_{-9}+x_{8} g_{-16}+x_{9} g_{-19}+x_{12} g_{-29}+x_{10} g_{-37}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(x_{3} x_{9} +x_{4} x_{10} +x_{6} x_{12} -14\right)h_{7}+\left(2x_{3} x_{9} +2x_{4} x_{10} +2x_{6} x_{12} -28\right)h_{6}+\left(2x_{2} x_{8} +2x_{4} x_{10} +2x_{6} x_{12} -26\right)h_{5}+\left(2x_{2} x_{8} +2x_{4} x_{10} -24\right)h_{4}+\left(2x_{2} x_{8} +2x_{5} x_{11} -22\right)h_{3}+\left(2x_{1} x_{7} +2x_{5} x_{11} -16\right)h_{2}+\left(2x_{1} x_{7} -10\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{7} -10&=&0\\2x_{1} x_{7} +2x_{5} x_{11} -16&=&0\\2x_{2} x_{8} +2x_{5} x_{11} -22&=&0\\2x_{2} x_{8} +2x_{4} x_{10} -24&=&0\\2x_{2} x_{8} +2x_{4} x_{10} +2x_{6} x_{12} -26&=&0\\2x_{3} x_{9} +2x_{4} x_{10} +2x_{6} x_{12} -28&=&0\\x_{3} x_{9} +x_{4} x_{10} +x_{6} x_{12} -14&=&0\\\end{array}

\begin{array}{rcl}h&=&14h_{7}+28h_{6}+26h_{5}+24h_{4}+22h_{3}+16h_{2}+10h_{1}\\e&=&x_{4} g_{37}+x_{6} g_{29}+x_{3} g_{19}+x_{2} g_{16}+x_{5} g_{9}+x_{1} g_{8}\\f&=&g_{-8}+g_{-9}+g_{-16}+g_{-19}+g_{-29}+g_{-37}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0\\ 2 & 0 & 0 & 0 & 2 & 0\\ 0 & 2 & 0 & 0 & 2 & 0\\ 0 & 2 & 0 & 2 & 0 & 0\\ 0 & 2 & 0 & 2 & 0 & 2\\ 0 & 0 & 2 & 2 & 0 & 2\\ 0 & 0 & 1 & 1 & 0 & 1\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}10\\ 16\\ 22\\ 24\\ 26\\ 28\\ 14\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&14h_{7}+28h_{6}+26h_{5}+24h_{4}+22h_{3}+16h_{2}+10h_{1}\\ e&=&x_{4} g_{37}+x_{6} g_{29}+x_{3} g_{19}+x_{2} g_{16}+x_{5} g_{9}+x_{1} g_{8}\\ f&=&x_{7} g_{-8}+x_{11} g_{-9}+x_{8} g_{-16}+x_{9} g_{-19}+x_{12} g_{-29}+x_{10} g_{-37}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{7} -10&=&0\\2x_{1} x_{7} +2x_{5} x_{11} -16&=&0\\2x_{2} x_{8} +2x_{5} x_{11} -22&=&0\\2x_{2} x_{8} +2x_{4} x_{10} -24&=&0\\2x_{2} x_{8} +2x_{4} x_{10} +2x_{6} x_{12} -26&=&0\\2x_{3} x_{9} +2x_{4} x_{10} +2x_{6} x_{12} -28&=&0\\x_{3} x_{9} +x_{4} x_{10} +x_{6} x_{12} -14&=&0\\\end{array}

A^{45}_1

h-characteristic: (2, 0, 2, 0, 1, 0, 0)
Length of the weight dual to h: 90
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1:

C^{1}_3+B^{1}_2

Containing regular semisimple subalgebra number 2:

C^{1}_5

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

V_{10\psi}+V_{8\psi}+3V_{6\psi}+4V_{5\psi}+V_{4\psi}+4V_{3\psi}+3V_{2\psi}+10V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&13h_{7}+26h_{6}+26h_{5}+24h_{4}+22h_{3}+16h_{2}+10h_{1}\\ e&=&4g_{37}+9g_{29}+8g_{10}+3g_{9}+5g_{8}\\ f&=&g_{-8}+g_{-9}+g_{-10}+g_{-29}+g_{-37}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&13h_{7}+26h_{6}+26h_{5}+24h_{4}+22h_{3}+16h_{2}+10h_{1}\\ [h, e]&=&8g_{37}+18g_{29}+16g_{10}+6g_{9}+10g_{8}\\ [h, f]&=&-2g_{-8}-2g_{-9}-2g_{-10}-2g_{-29}-2g_{-37}\end{array}

Centralizer type:

B_2

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&13h_{7}+26h_{6}+26h_{5}+24h_{4}+22h_{3}+16h_{2}+10h_{1}\\ e&=&x_{4} g_{37}+x_{3} g_{29}+x_{2} g_{10}+x_{5} g_{9}+x_{1} g_{8}\\ f&=&x_{6} g_{-8}+x_{10} g_{-9}+x_{7} g_{-10}+x_{8} g_{-29}+x_{9} g_{-37}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(x_{3} x_{8} +x_{4} x_{9} -13\right)h_{7}+\left(2x_{3} x_{8} +2x_{4} x_{9} -26\right)h_{6}+\left(2x_{3} x_{8} +2x_{4} x_{9} -26\right)h_{5}+\left(2x_{2} x_{7} +2x_{4} x_{9} -24\right)h_{4}+\left(2x_{2} x_{7} +2x_{5} x_{10} -22\right)h_{3}+\left(2x_{1} x_{6} +2x_{5} x_{10} -16\right)h_{2}+\left(2x_{1} x_{6} -10\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{6} -10&=&0\\2x_{1} x_{6} +2x_{5} x_{10} -16&=&0\\2x_{2} x_{7} +2x_{5} x_{10} -22&=&0\\2x_{2} x_{7} +2x_{4} x_{9} -24&=&0\\2x_{3} x_{8} +2x_{4} x_{9} -26&=&0\\2x_{3} x_{8} +2x_{4} x_{9} -26&=&0\\x_{3} x_{8} +x_{4} x_{9} -13&=&0\\\end{array}

\begin{array}{rcl}h&=&13h_{7}+26h_{6}+26h_{5}+24h_{4}+22h_{3}+16h_{2}+10h_{1}\\e&=&x_{4} g_{37}+x_{3} g_{29}+x_{2} g_{10}+x_{5} g_{9}+x_{1} g_{8}\\f&=&g_{-8}+g_{-9}+g_{-10}+g_{-29}+g_{-37}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0 & 0 & 0\\ 2 & 0 & 0 & 0 & 2\\ 0 & 2 & 0 & 0 & 2\\ 0 & 2 & 0 & 2 & 0\\ 0 & 0 & 2 & 2 & 0\\ 0 & 0 & 2 & 2 & 0\\ 0 & 0 & 1 & 1 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}10\\ 16\\ 22\\ 24\\ 26\\ 26\\ 13\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&13h_{7}+26h_{6}+26h_{5}+24h_{4}+22h_{3}+16h_{2}+10h_{1}\\ e&=&x_{4} g_{37}+x_{3} g_{29}+x_{2} g_{10}+x_{5} g_{9}+x_{1} g_{8}\\ f&=&x_{6} g_{-8}+x_{10} g_{-9}+x_{7} g_{-10}+x_{8} g_{-29}+x_{9} g_{-37}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{6} -10&=&0\\2x_{1} x_{6} +2x_{5} x_{10} -16&=&0\\2x_{2} x_{7} +2x_{5} x_{10} -22&=&0\\2x_{2} x_{7} +2x_{4} x_{9} -24&=&0\\2x_{3} x_{8} +2x_{4} x_{9} -26&=&0\\2x_{3} x_{8} +2x_{4} x_{9} -26&=&0\\x_{3} x_{8} +x_{4} x_{9} -13&=&0\\\end{array}

A^{44}_1

h-characteristic: (2, 1, 0, 1, 0, 1, 0)
Length of the weight dual to h: 88
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1:

C^{1}_3+A^{2}_2+A^{1}_1

Containing regular semisimple subalgebra number 2:

C^{1}_4+A^{2}_2

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

V_{10\psi}+2V_{7\psi}+2V_{6\psi}+2V_{5\psi}+4V_{4\psi}+4V_{3\psi}+3V_{2\psi}+2V_{\psi}+3V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&14h_{7}+28h_{6}+26h_{5}+24h_{4}+20h_{3}+16h_{2}+10h_{1}\\ e&=&g_{29}+2g_{27}+8g_{21}+9g_{19}+2g_{17}+5g_{1}\\ f&=&g_{-1}+g_{-17}+g_{-19}+g_{-21}+g_{-27}+g_{-29}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&14h_{7}+28h_{6}+26h_{5}+24h_{4}+20h_{3}+16h_{2}+10h_{1}\\ [h, e]&=&2g_{29}+4g_{27}+16g_{21}+18g_{19}+4g_{17}+10g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-17}-2g_{-19}-2g_{-21}-2g_{-27}-2g_{-29}\end{array}

Centralizer type:

A^{3}_1

Killing form square of Cartan element dual to ambient long root: 32
Basis of the centralizer (dimension: 3):

h_{7}-2h_{3}

g_{3}+g_{-7}

g_{7}+g_{-3}

Basis of centralizer intersected with cartan (dimension: 1):

1/2h_{7}-h_{3}

Cartan of centralizer (dimension: 1):

1/2h_{7}-h_{3}

Cartan-generating semisimple element:

1/2h_{7}-h_{3}

adjoint action:

\begin{pmatrix}0 & 0 & 0\\ 0 & -1 & 0\\ 0 & 0 & 1\\ \end{pmatrix}

Characteristic polynomial ad H:

x^3-x

Factorization of characteristic polynomial of ad H: (x )(x -1)(x +1)
Eigenvalues of ad H:

0

1

-1

3 eigenvectors of ad H:

(1,0,0)

(0,0,1)

(0,1,0)

Centralizer type: A^{3}_1
Reductive components (1 total):
Scalar product computed:

\begin{pmatrix}1/24\\ \end{pmatrix}

Simple basis of Cartan of centralizer (1 total):

h_{7}-2h_{3}

matching e:

g_{7}+g_{-3}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0\\ 0 & -2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}

Linear space basis of intersection of centralizer and ambient Cartan:

h_{7}-2h_{3}

matching e:

g_{7}+g_{-3}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0\\ 0 & -2 & 0\\ 0 & 0 & 2\\ \end{pmatrix}

Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form:

\begin{pmatrix}96\\ \end{pmatrix}

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&14h_{7}+28h_{6}+26h_{5}+24h_{4}+20h_{3}+16h_{2}+10h_{1}\\ e&=&x_{6} g_{29}+x_{4} g_{27}+x_{2} g_{21}+x_{3} g_{19}+x_{5} g_{17}+x_{1} g_{1}\\ f&=&x_{7} g_{-1}+x_{11} g_{-17}+x_{9} g_{-19}+x_{8} g_{-21}+x_{10} g_{-27}+x_{12} g_{-29}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(x_{3} x_{9} +2x_{4} x_{10} +x_{6} x_{12} -14\right)h_{7}+\left(2x_{3} x_{9} +2x_{4} x_{10} +2x_{5} x_{11} +2x_{6} x_{12} -28\right)h_{6}+\left(2x_{2} x_{8} +2x_{4} x_{10} +2x_{5} x_{11} +2x_{6} x_{12} -26\right)h_{5}+\left(2x_{2} x_{8} +2x_{4} x_{10} +2x_{5} x_{11} -24\right)h_{4}+\left(2x_{2} x_{8} +2x_{4} x_{10} -20\right)h_{3}+\left(2x_{2} x_{8} -16\right)h_{2}+\left(2x_{1} x_{7} -10\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{7} -10&=&0\\2x_{2} x_{8} -16&=&0\\2x_{2} x_{8} +2x_{4} x_{10} -20&=&0\\2x_{2} x_{8} +2x_{4} x_{10} +2x_{5} x_{11} -24&=&0\\2x_{2} x_{8} +2x_{4} x_{10} +2x_{5} x_{11} +2x_{6} x_{12} -26&=&0\\2x_{3} x_{9} +2x_{4} x_{10} +2x_{5} x_{11} +2x_{6} x_{12} -28&=&0\\x_{3} x_{9} +2x_{4} x_{10} +x_{6} x_{12} -14&=&0\\\end{array}

\begin{array}{rcl}h&=&14h_{7}+28h_{6}+26h_{5}+24h_{4}+20h_{3}+16h_{2}+10h_{1}\\e&=&x_{6} g_{29}+x_{4} g_{27}+x_{2} g_{21}+x_{3} g_{19}+x_{5} g_{17}+x_{1} g_{1}\\f&=&g_{-1}+g_{-17}+g_{-19}+g_{-21}+g_{-27}+g_{-29}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 2 & 0 & 0\\ 0 & 2 & 0 & 2 & 2 & 0\\ 0 & 2 & 0 & 2 & 2 & 2\\ 0 & 0 & 2 & 2 & 2 & 2\\ 0 & 0 & 1 & 2 & 0 & 1\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}10\\ 16\\ 20\\ 24\\ 26\\ 28\\ 14\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&14h_{7}+28h_{6}+26h_{5}+24h_{4}+20h_{3}+16h_{2}+10h_{1}\\ e&=&x_{6} g_{29}+x_{4} g_{27}+x_{2} g_{21}+x_{3} g_{19}+x_{5} g_{17}+x_{1} g_{1}\\ f&=&x_{7} g_{-1}+x_{11} g_{-17}+x_{9} g_{-19}+x_{8} g_{-21}+x_{10} g_{-27}+x_{12} g_{-29}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{7} -10&=&0\\2x_{2} x_{8} -16&=&0\\2x_{2} x_{8} +2x_{4} x_{10} -20&=&0\\2x_{2} x_{8} +2x_{4} x_{10} +2x_{5} x_{11} -24&=&0\\2x_{2} x_{8} +2x_{4} x_{10} +2x_{5} x_{11} +2x_{6} x_{12} -26&=&0\\2x_{3} x_{9} +2x_{4} x_{10} +2x_{5} x_{11} +2x_{6} x_{12} -28&=&0\\x_{3} x_{9} +2x_{4} x_{10} +x_{6} x_{12} -14&=&0\\\end{array}

A^{43}_1

h-characteristic: (2, 1, 0, 1, 1, 0, 0)
Length of the weight dual to h: 86
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1:

C^{1}_3+A^{2}_2

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

V_{10\psi}+2V_{7\psi}+V_{6\psi}+4V_{5\psi}+3V_{4\psi}+2V_{3\psi}+6V_{2\psi}+6V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&13h_{7}+26h_{6}+26h_{5}+24h_{4}+20h_{3}+16h_{2}+10h_{1}\\ e&=&2g_{32}+9g_{29}+8g_{15}+2g_{11}+5g_{1}\\ f&=&g_{-1}+g_{-11}+g_{-15}+g_{-29}+g_{-32}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&13h_{7}+26h_{6}+26h_{5}+24h_{4}+20h_{3}+16h_{2}+10h_{1}\\ [h, e]&=&4g_{32}+18g_{29}+16g_{15}+4g_{11}+10g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-11}-2g_{-15}-2g_{-29}-2g_{-32}\end{array}

Centralizer type:

A^{3}_1+A_1

Killing form square of Cartan element dual to ambient long root: 32
Basis of the centralizer (dimension: 6):

g_{-7}

h_{6}-h_{3}

h_{7}

g_{3}+g_{-19}

g_{7}

g_{19}+g_{-3}

Basis of centralizer intersected with cartan (dimension: 2):

h_{6}-h_{3}

-h_{7}

Cartan of centralizer (dimension: 2):

h_{6}-h_{3}

-h_{7}

Cartan-generating semisimple element:

-h_{7}+h_{6}-h_{3}

adjoint action:

\begin{pmatrix}3 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & -3 & 0\\ 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}

Characteristic polynomial ad H:

x^6-10x^4+9x^2

Factorization of characteristic polynomial of ad H: (x )(x )(x -3)(x -1)(x +1)(x +3)
Eigenvalues of ad H:

0

3

1

-1

-3

6 eigenvectors of ad H:

(0,1,0,0,0,0)

(0,0,1,0,0,0)

(1,0,0,0,0,0)

(0,0,0,0,0,1)

(0,0,0,1,0,0)

(0,0,0,0,1,0)

Centralizer type: A^{3}_1+A^{1}_1
Reductive components (2 total):
Scalar product computed:

\begin{pmatrix}1/8\\ \end{pmatrix}

Simple basis of Cartan of centralizer (1 total):

-h_{7}

matching e:

g_{-7}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

Linear space basis of intersection of centralizer and ambient Cartan:

-h_{7}

matching e:

g_{-7}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form:

\begin{pmatrix}32\\ \end{pmatrix}

Scalar product computed:

\begin{pmatrix}1/24\\ \end{pmatrix}

Simple basis of Cartan of centralizer (1 total):

h_{7}+2h_{6}-2h_{3}

matching e:

g_{19}+g_{-3}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}

Linear space basis of intersection of centralizer and ambient Cartan:

h_{7}+2h_{6}-2h_{3}

matching e:

g_{19}+g_{-3}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}

Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form:

\begin{pmatrix}96\\ \end{pmatrix}

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&13h_{7}+26h_{6}+26h_{5}+24h_{4}+20h_{3}+16h_{2}+10h_{1}\\ e&=&x_{4} g_{32}+x_{3} g_{29}+x_{2} g_{15}+x_{5} g_{11}+x_{1} g_{1}\\ f&=&x_{6} g_{-1}+x_{10} g_{-11}+x_{7} g_{-15}+x_{8} g_{-29}+x_{9} g_{-32}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(x_{3} x_{8} +2x_{4} x_{9} -13\right)h_{7}+\left(2x_{3} x_{8} +4x_{4} x_{9} -26\right)h_{6}+\left(2x_{3} x_{8} +2x_{4} x_{9} +2x_{5} x_{10} -26\right)h_{5}+\left(2x_{2} x_{7} +2x_{4} x_{9} +2x_{5} x_{10} -24\right)h_{4}+\left(2x_{2} x_{7} +2x_{4} x_{9} -20\right)h_{3}+\left(2x_{2} x_{7} -16\right)h_{2}+\left(2x_{1} x_{6} -10\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{6} -10&=&0\\2x_{2} x_{7} -16&=&0\\2x_{2} x_{7} +2x_{4} x_{9} -20&=&0\\2x_{2} x_{7} +2x_{4} x_{9} +2x_{5} x_{10} -24&=&0\\2x_{3} x_{8} +2x_{4} x_{9} +2x_{5} x_{10} -26&=&0\\2x_{3} x_{8} +4x_{4} x_{9} -26&=&0\\x_{3} x_{8} +2x_{4} x_{9} -13&=&0\\\end{array}

\begin{array}{rcl}h&=&13h_{7}+26h_{6}+26h_{5}+24h_{4}+20h_{3}+16h_{2}+10h_{1}\\e&=&x_{4} g_{32}+x_{3} g_{29}+x_{2} g_{15}+x_{5} g_{11}+x_{1} g_{1}\\f&=&g_{-1}+g_{-11}+g_{-15}+g_{-29}+g_{-32}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0\\ 0 & 2 & 0 & 2 & 0\\ 0 & 2 & 0 & 2 & 2\\ 0 & 0 & 2 & 2 & 2\\ 0 & 0 & 2 & 4 & 0\\ 0 & 0 & 1 & 2 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}10\\ 16\\ 20\\ 24\\ 26\\ 26\\ 13\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&13h_{7}+26h_{6}+26h_{5}+24h_{4}+20h_{3}+16h_{2}+10h_{1}\\ e&=&x_{4} g_{32}+x_{3} g_{29}+x_{2} g_{15}+x_{5} g_{11}+x_{1} g_{1}\\ f&=&x_{6} g_{-1}+x_{10} g_{-11}+x_{7} g_{-15}+x_{8} g_{-29}+x_{9} g_{-32}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{6} -10&=&0\\2x_{2} x_{7} -16&=&0\\2x_{2} x_{7} +2x_{4} x_{9} -20&=&0\\2x_{2} x_{7} +2x_{4} x_{9} +2x_{5} x_{10} -24&=&0\\2x_{3} x_{8} +2x_{4} x_{9} +2x_{5} x_{10} -26&=&0\\2x_{3} x_{8} +4x_{4} x_{9} -26&=&0\\x_{3} x_{8} +2x_{4} x_{9} -13&=&0\\\end{array}

A^{42}_1

h-characteristic: (0, 2, 0, 1, 0, 1, 0)
Length of the weight dual to h: 84
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1:

A^{2}_4+2A^{1}_1

Containing regular semisimple subalgebra number 2:

A^{2}_4+A^{2}_1

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

3V_{8\psi}+V_{6\psi}+4V_{5\psi}+3V_{4\psi}+4V_{3\psi}+4V_{2\psi}+4V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&14h_{7}+28h_{6}+26h_{5}+24h_{4}+20h_{3}+16h_{2}+8h_{1}\\ e&=&g_{29}+6g_{23}+6g_{22}+g_{19}+4g_{14}+4g_{2}\\ f&=&g_{-2}+g_{-14}+g_{-19}+g_{-22}+g_{-23}+g_{-29}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&14h_{7}+28h_{6}+26h_{5}+24h_{4}+20h_{3}+16h_{2}+8h_{1}\\ [h, e]&=&2g_{29}+12g_{23}+12g_{22}+2g_{19}+8g_{14}+8g_{2}\\ [h, f]&=&-2g_{-2}-2g_{-14}-2g_{-19}-2g_{-22}-2g_{-23}-2g_{-29}\end{array}

Centralizer type:

A^{5}_1

Killing form square of Cartan element dual to ambient long root: 32
Basis of the centralizer (dimension: 4):

h_{7}+2h_{3}-2h_{1}

g_{1}+g_{-3}+g_{-7}

g_{5}-g_{-5}

g_{7}+g_{3}+g_{-1}

Basis of centralizer intersected with cartan (dimension: 1):

1/2h_{7}+h_{3}-h_{1}

Cartan of centralizer (dimension: 2):

1/2h_{7}+h_{3}-h_{1}

g_{5}-g_{-5}

Cartan-generating semisimple element:

1/2h_{7}+h_{3}-h_{1}

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0\\ 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\\ \end{pmatrix}

Characteristic polynomial ad H:

x^4-x^2

Factorization of characteristic polynomial of ad H: (x )(x )(x -1)(x +1)
Eigenvalues of ad H:

0

1

-1

4 eigenvectors of ad H:

(1,0,0,0)

(0,0,1,0)

(0,0,0,1)

(0,1,0,0)

Centralizer type: A^{5}_1
Reductive components (1 total):
Scalar product computed:

\begin{pmatrix}1/40\\ \end{pmatrix}

Simple basis of Cartan of centralizer (1 total):

h_{7}+2h_{3}-2h_{1}

matching e:

g_{7}+g_{3}+g_{-1}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0\\ 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2\\ \end{pmatrix}

Linear space basis of intersection of centralizer and ambient Cartan:

h_{7}+2h_{3}-2h_{1}

matching e:

g_{7}+g_{3}+g_{-1}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0\\ 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2\\ \end{pmatrix}

Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form:

\begin{pmatrix}160\\ \end{pmatrix}

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&14h_{7}+28h_{6}+26h_{5}+24h_{4}+20h_{3}+16h_{2}+8h_{1}\\ e&=&x_{5} g_{29}+x_{2} g_{23}+x_{3} g_{22}+x_{6} g_{19}+x_{1} g_{14}+x_{4} g_{2}\\ f&=&x_{10} g_{-2}+x_{7} g_{-14}+x_{12} g_{-19}+x_{9} g_{-22}+x_{8} g_{-23}+x_{11} g_{-29}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(2x_{2} x_{8} +x_{5} x_{11} +x_{6} x_{12} -14\right)h_{7}+\left(2x_{2} x_{8} +2x_{3} x_{9} +2x_{5} x_{11} +2x_{6} x_{12} -28\right)h_{6}+\left(2x_{2} x_{8} +2x_{3} x_{9} +2x_{5} x_{11} -26\right)h_{5}+\left(2x_{2} x_{8} +2x_{3} x_{9} -24\right)h_{4}+\left(2x_{1} x_{7} +2x_{3} x_{9} -20\right)h_{3}+\left(2x_{1} x_{7} +2x_{4} x_{10} -16\right)h_{2}+\left(2x_{1} x_{7} -8\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{7} -8&=&0\\2x_{1} x_{7} +2x_{4} x_{10} -16&=&0\\2x_{1} x_{7} +2x_{3} x_{9} -20&=&0\\2x_{2} x_{8} +2x_{3} x_{9} -24&=&0\\2x_{2} x_{8} +2x_{3} x_{9} +2x_{5} x_{11} -26&=&0\\2x_{2} x_{8} +2x_{3} x_{9} +2x_{5} x_{11} +2x_{6} x_{12} -28&=&0\\2x_{2} x_{8} +x_{5} x_{11} +x_{6} x_{12} -14&=&0\\\end{array}

\begin{array}{rcl}h&=&14h_{7}+28h_{6}+26h_{5}+24h_{4}+20h_{3}+16h_{2}+8h_{1}\\e&=&x_{5} g_{29}+x_{2} g_{23}+x_{3} g_{22}+x_{6} g_{19}+x_{1} g_{14}+x_{4} g_{2}\\f&=&g_{-2}+g_{-14}+g_{-19}+g_{-22}+g_{-23}+g_{-29}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0\\ 2 & 0 & 0 & 2 & 0 & 0\\ 2 & 0 & 2 & 0 & 0 & 0\\ 0 & 2 & 2 & 0 & 0 & 0\\ 0 & 2 & 2 & 0 & 2 & 0\\ 0 & 2 & 2 & 0 & 2 & 2\\ 0 & 2 & 0 & 0 & 1 & 1\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}8\\ 16\\ 20\\ 24\\ 26\\ 28\\ 14\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&14h_{7}+28h_{6}+26h_{5}+24h_{4}+20h_{3}+16h_{2}+8h_{1}\\ e&=&x_{5} g_{29}+x_{2} g_{23}+x_{3} g_{22}+x_{6} g_{19}+x_{1} g_{14}+x_{4} g_{2}\\ f&=&x_{10} g_{-2}+x_{7} g_{-14}+x_{12} g_{-19}+x_{9} g_{-22}+x_{8} g_{-23}+x_{11} g_{-29}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{7} -8&=&0\\2x_{1} x_{7} +2x_{4} x_{10} -16&=&0\\2x_{1} x_{7} +2x_{3} x_{9} -20&=&0\\2x_{2} x_{8} +2x_{3} x_{9} -24&=&0\\2x_{2} x_{8} +2x_{3} x_{9} +2x_{5} x_{11} -26&=&0\\2x_{2} x_{8} +2x_{3} x_{9} +2x_{5} x_{11} +2x_{6} x_{12} -28&=&0\\2x_{2} x_{8} +x_{5} x_{11} +x_{6} x_{12} -14&=&0\\\end{array}

A^{41}_1

h-characteristic: (0, 2, 0, 1, 1, 0, 0)
Length of the weight dual to h: 82
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1:

A^{2}_4+A^{1}_1

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

3V_{8\psi}+V_{6\psi}+2V_{5\psi}+7V_{4\psi}+2V_{3\psi}+2V_{2\psi}+2V_{\psi}+6V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&13h_{7}+26h_{6}+26h_{5}+24h_{4}+20h_{3}+16h_{2}+8h_{1}\\ e&=&g_{29}+6g_{28}+6g_{16}+4g_{14}+4g_{2}\\ f&=&g_{-2}+g_{-14}+g_{-16}+g_{-28}+g_{-29}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&13h_{7}+26h_{6}+26h_{5}+24h_{4}+20h_{3}+16h_{2}+8h_{1}\\ [h, e]&=&2g_{29}+12g_{28}+12g_{16}+8g_{14}+8g_{2}\\ [h, f]&=&-2g_{-2}-2g_{-14}-2g_{-16}-2g_{-28}-2g_{-29}\end{array}

Centralizer type:

A^{5}_1+A_1

Killing form square of Cartan element dual to ambient long root: 32
Basis of the centralizer (dimension: 6):

g_{-7}

h_{6}+h_{3}-h_{1}

h_{7}

g_{1}+g_{-3}+g_{-19}

g_{7}

g_{19}+g_{3}+g_{-1}

Basis of centralizer intersected with cartan (dimension: 2):

h_{6}+h_{3}-h_{1}

-h_{7}

Cartan of centralizer (dimension: 2):

h_{6}+h_{3}-h_{1}

-h_{7}

Cartan-generating semisimple element:

-h_{7}+h_{6}+h_{3}-h_{1}

adjoint action:

\begin{pmatrix}3 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & -3 & 0\\ 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}

Characteristic polynomial ad H:

x^6-10x^4+9x^2

Factorization of characteristic polynomial of ad H: (x )(x )(x -3)(x -1)(x +1)(x +3)
Eigenvalues of ad H:

0

3

1

-1

-3

6 eigenvectors of ad H:

(0,1,0,0,0,0)

(0,0,1,0,0,0)

(1,0,0,0,0,0)

(0,0,0,0,0,1)

(0,0,0,1,0,0)

(0,0,0,0,1,0)

Centralizer type: A^{5}_1+A^{1}_1
Reductive components (2 total):
Scalar product computed:

\begin{pmatrix}1/8\\ \end{pmatrix}

Simple basis of Cartan of centralizer (1 total):

-h_{7}

matching e:

g_{-7}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

Linear space basis of intersection of centralizer and ambient Cartan:

-h_{7}

matching e:

g_{-7}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form:

\begin{pmatrix}32\\ \end{pmatrix}

Scalar product computed:

\begin{pmatrix}1/40\\ \end{pmatrix}

Simple basis of Cartan of centralizer (1 total):

h_{7}+2h_{6}+2h_{3}-2h_{1}

matching e:

g_{19}+g_{3}+g_{-1}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}

Linear space basis of intersection of centralizer and ambient Cartan:

h_{7}+2h_{6}+2h_{3}-2h_{1}

matching e:

g_{19}+g_{3}+g_{-1}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}

Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form:

\begin{pmatrix}160\\ \end{pmatrix}

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&13h_{7}+26h_{6}+26h_{5}+24h_{4}+20h_{3}+16h_{2}+8h_{1}\\ e&=&x_{5} g_{29}+x_{2} g_{28}+x_{3} g_{16}+x_{1} g_{14}+x_{4} g_{2}\\ f&=&x_{9} g_{-2}+x_{6} g_{-14}+x_{8} g_{-16}+x_{7} g_{-28}+x_{10} g_{-29}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(2x_{2} x_{7} +x_{5} x_{10} -13\right)h_{7}+\left(4x_{2} x_{7} +2x_{5} x_{10} -26\right)h_{6}+\left(2x_{2} x_{7} +2x_{3} x_{8} +2x_{5} x_{10} -26\right)h_{5}+\left(2x_{2} x_{7} +2x_{3} x_{8} -24\right)h_{4}+\left(2x_{1} x_{6} +2x_{3} x_{8} -20\right)h_{3}+\left(2x_{1} x_{6} +2x_{4} x_{9} -16\right)h_{2}+\left(2x_{1} x_{6} -8\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{6} -8&=&0\\2x_{1} x_{6} +2x_{4} x_{9} -16&=&0\\2x_{1} x_{6} +2x_{3} x_{8} -20&=&0\\2x_{2} x_{7} +2x_{3} x_{8} -24&=&0\\2x_{2} x_{7} +2x_{3} x_{8} +2x_{5} x_{10} -26&=&0\\4x_{2} x_{7} +2x_{5} x_{10} -26&=&0\\2x_{2} x_{7} +x_{5} x_{10} -13&=&0\\\end{array}

\begin{array}{rcl}h&=&13h_{7}+26h_{6}+26h_{5}+24h_{4}+20h_{3}+16h_{2}+8h_{1}\\e&=&x_{5} g_{29}+x_{2} g_{28}+x_{3} g_{16}+x_{1} g_{14}+x_{4} g_{2}\\f&=&g_{-2}+g_{-14}+g_{-16}+g_{-28}+g_{-29}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0 & 0 & 0\\ 2 & 0 & 0 & 2 & 0\\ 2 & 0 & 2 & 0 & 0\\ 0 & 2 & 2 & 0 & 0\\ 0 & 2 & 2 & 0 & 2\\ 0 & 4 & 0 & 0 & 2\\ 0 & 2 & 0 & 0 & 1\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}8\\ 16\\ 20\\ 24\\ 26\\ 26\\ 13\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&13h_{7}+26h_{6}+26h_{5}+24h_{4}+20h_{3}+16h_{2}+8h_{1}\\ e&=&x_{5} g_{29}+x_{2} g_{28}+x_{3} g_{16}+x_{1} g_{14}+x_{4} g_{2}\\ f&=&x_{9} g_{-2}+x_{6} g_{-14}+x_{8} g_{-16}+x_{7} g_{-28}+x_{10} g_{-29}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{6} -8&=&0\\2x_{1} x_{6} +2x_{4} x_{9} -16&=&0\\2x_{1} x_{6} +2x_{3} x_{8} -20&=&0\\2x_{2} x_{7} +2x_{3} x_{8} -24&=&0\\2x_{2} x_{7} +2x_{3} x_{8} +2x_{5} x_{10} -26&=&0\\4x_{2} x_{7} +2x_{5} x_{10} -26&=&0\\2x_{2} x_{7} +x_{5} x_{10} -13&=&0\\\end{array}

A^{40}_1

h-characteristic: (0, 2, 0, 2, 0, 0, 0)
Length of the weight dual to h: 80
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1:

A^{2}_4

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

3V_{8\psi}+V_{6\psi}+11V_{4\psi}+V_{2\psi}+13V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&12h_{7}+24h_{6}+24h_{5}+24h_{4}+20h_{3}+16h_{2}+8h_{1}\\ e&=&6g_{33}+4g_{14}+6g_{10}+4g_{2}\\ f&=&g_{-2}+g_{-10}+g_{-14}+g_{-33}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&12h_{7}+24h_{6}+24h_{5}+24h_{4}+20h_{3}+16h_{2}+8h_{1}\\ [h, e]&=&12g_{33}+8g_{14}+12g_{10}+8g_{2}\\ [h, f]&=&-2g_{-2}-2g_{-10}-2g_{-14}-2g_{-33}\end{array}

Centralizer type:

B_2+A^{5}_1

Killing form square of Cartan element dual to ambient long root: 32
Basis of the centralizer (dimension: 13):

g_{-19}

g_{-13}

g_{-7}

g_{-6}

h_{5}+h_{3}-h_{1}

h_{6}

h_{7}

g_{1}+g_{-3}+g_{-29}

g_{6}

g_{7}

g_{13}

g_{19}

g_{29}+g_{3}+g_{-1}

Basis of centralizer intersected with cartan (dimension: 3):

h_{5}+h_{3}-h_{1}

-h_{7}

-h_{6}

Cartan of centralizer (dimension: 3):

-h_{6}

-h_{7}

h_{5}+h_{3}-h_{1}

Cartan-generating semisimple element:

-h_{7}-h_{6}+h_{5}+h_{3}-h_{1}

adjoint action:

\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 3/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1/2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -3/2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}

Characteristic polynomial ad H:

x^{13}-17/2x^{11}+393/16x^9-239/8x^7+241/16x^5-9/4x^3

Factorization of characteristic polynomial of ad H: 1/16(x )(x )(x )(x -2)(x -1)(x -1)(x +1)(x +1)(x +2)(2x -3)(2x -1)(2x +1)(2x +3)
Eigenvalues of ad H:

0

2

1

-1

-2

3/2

1/2

-1/2

-3/2

13 eigenvectors of ad H:

(0,0,0,0,1,0,0,0,0,0,0,0,0)

(0,0,0,0,0,1,0,0,0,0,0,0,0)

(0,0,0,0,0,0,1,0,0,0,0,0,0)

(1,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,1,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,1)

(0,0,0,0,0,0,0,1,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,1,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,1,0)

(0,1,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,1,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,1,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,1,0,0)

Centralizer type: B^{1}_2+A^{5}_1
Reductive components (2 total):
Scalar product computed:

\begin{pmatrix}1/8 & -1/16\\ -1/16 & 1/16\\ \end{pmatrix}

Simple basis of Cartan of centralizer (2 total):

-h_{7}

matching e:

g_{-7}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

-2h_{6}

matching e:

g_{-6}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

Linear space basis of intersection of centralizer and ambient Cartan:

-h_{7}

matching e:

g_{-7}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

-2h_{6}

matching e:

g_{-6}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

Elements in Cartan dual to root system: (1, 1), (-1, -1), (1, 0), (-1, 0), (2, 1), (-2, -1), (0, 1), (0, -1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form:

\begin{pmatrix}32 & -32\\ -32 & 64\\ \end{pmatrix}

Scalar product computed:

\begin{pmatrix}1/40\\ \end{pmatrix}

Simple basis of Cartan of centralizer (1 total):

h_{7}+2h_{6}+2h_{5}+2h_{3}-2h_{1}

matching e:

g_{29}+g_{3}+g_{-1}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}

Linear space basis of intersection of centralizer and ambient Cartan:

h_{7}+2h_{6}+2h_{5}+2h_{3}-2h_{1}

matching e:

g_{29}+g_{3}+g_{-1}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}

Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form:

\begin{pmatrix}160\\ \end{pmatrix}

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&12h_{7}+24h_{6}+24h_{5}+24h_{4}+20h_{3}+16h_{2}+8h_{1}\\ e&=&x_{2} g_{33}+x_{1} g_{14}+x_{3} g_{10}+x_{4} g_{2}\\ f&=&x_{8} g_{-2}+x_{7} g_{-10}+x_{5} g_{-14}+x_{6} g_{-33}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(2x_{2} x_{6} -12\right)h_{7}+\left(4x_{2} x_{6} -24\right)h_{6}+\left(4x_{2} x_{6} -24\right)h_{5}+\left(2x_{2} x_{6} +2x_{3} x_{7} -24\right)h_{4}+\left(2x_{1} x_{5} +2x_{3} x_{7} -20\right)h_{3}+\left(2x_{1} x_{5} +2x_{4} x_{8} -16\right)h_{2}+\left(2x_{1} x_{5} -8\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{5} -8&=&0\\2x_{1} x_{5} +2x_{4} x_{8} -16&=&0\\2x_{1} x_{5} +2x_{3} x_{7} -20&=&0\\2x_{2} x_{6} +2x_{3} x_{7} -24&=&0\\4x_{2} x_{6} -24&=&0\\4x_{2} x_{6} -24&=&0\\2x_{2} x_{6} -12&=&0\\\end{array}

\begin{array}{rcl}h&=&12h_{7}+24h_{6}+24h_{5}+24h_{4}+20h_{3}+16h_{2}+8h_{1}\\e&=&x_{2} g_{33}+x_{1} g_{14}+x_{3} g_{10}+x_{4} g_{2}\\f&=&g_{-2}+g_{-10}+g_{-14}+g_{-33}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0 & 0\\ 2 & 0 & 0 & 2\\ 2 & 0 & 2 & 0\\ 0 & 2 & 2 & 0\\ 0 & 4 & 0 & 0\\ 0 & 4 & 0 & 0\\ 0 & 2 & 0 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}8\\ 16\\ 20\\ 24\\ 24\\ 24\\ 12\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&12h_{7}+24h_{6}+24h_{5}+24h_{4}+20h_{3}+16h_{2}+8h_{1}\\ e&=&x_{2} g_{33}+x_{1} g_{14}+x_{3} g_{10}+x_{4} g_{2}\\ f&=&x_{8} g_{-2}+x_{7} g_{-10}+x_{5} g_{-14}+x_{6} g_{-33}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{5} -8&=&0\\2x_{1} x_{5} +2x_{4} x_{8} -16&=&0\\2x_{1} x_{5} +2x_{3} x_{7} -20&=&0\\2x_{2} x_{6} +2x_{3} x_{7} -24&=&0\\4x_{2} x_{6} -24&=&0\\4x_{2} x_{6} -24&=&0\\2x_{2} x_{6} -12&=&0\\\end{array}

A^{39}_1

h-characteristic: (2, 2, 0, 0, 0, 0, 2)
Length of the weight dual to h: 78
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 5
Containing regular semisimple subalgebra number 1:

C^{1}_3+4A^{1}_1

Containing regular semisimple subalgebra number 2:

C^{1}_3+A^{2}_1+2A^{1}_1

Containing regular semisimple subalgebra number 3:

C^{1}_3+2A^{2}_1

Containing regular semisimple subalgebra number 4:

C^{1}_4+3A^{1}_1

Containing regular semisimple subalgebra number 5:

C^{1}_4+A^{2}_1+A^{1}_1

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

V_{10\psi}+5V_{6\psi}+4V_{4\psi}+11V_{2\psi}+6V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&13h_{7}+24h_{6}+22h_{5}+20h_{4}+18h_{3}+16h_{2}+10h_{1}\\ e&=&g_{43}+g_{37}+g_{29}+8g_{26}+g_{19}+9g_{7}+5g_{1}\\ f&=&g_{-1}+g_{-7}+g_{-19}+g_{-26}+g_{-29}+g_{-37}+g_{-43}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&13h_{7}+24h_{6}+22h_{5}+20h_{4}+18h_{3}+16h_{2}+10h_{1}\\ [h, e]&=&2g_{43}+2g_{37}+2g_{29}+16g_{26}+2g_{19}+18g_{7}+10g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-7}-2g_{-19}-2g_{-26}-2g_{-29}-2g_{-37}-2g_{-43}\end{array}

Centralizer type:

2A^{4}_1

Killing form square of Cartan element dual to ambient long root: 32
Basis of the centralizer (dimension: 6):

g_{3}-g_{-3}

g_{4}-g_{-4}

g_{5}-g_{-5}

g_{10}-g_{-10}

g_{11}-g_{-11}

g_{16}-g_{-16}

Basis of centralizer intersected with cartan (dimension: 0):
Cartan of centralizer (dimension: 2):

g_{16}+g_{5}+g_{4}+g_{3}-g_{-3}-g_{-4}-g_{-5}-g_{-16}

g_{11}+g_{10}-g_{-10}-g_{-11}

Cartan-generating semisimple element:

g_{16}+g_{11}+g_{10}+g_{5}+g_{4}+g_{3}-g_{-3}-g_{-4}-g_{-5}-g_{-10}-g_{-11}-g_{-16}

adjoint action:

\begin{pmatrix}0 & -1 & 0 & 1 & -1 & 1\\ 1 & 0 & -1 & -1 & 1 & 0\\ 0 & 1 & 0 & 1 & -1 & -1\\ -1 & 1 & -1 & 0 & 0 & 1\\ 1 & -1 & 1 & 0 & 0 & -1\\ -1 & 0 & 1 & -1 & 1 & 0\\ \end{pmatrix}

Characteristic polynomial ad H:

x^6+12x^4+32x^2

Factorization of characteristic polynomial of ad H: (x )(x )(x^2+4)(x^2+8)
Eigenvalues of ad H:

0

2\sqrt{-1}

-2\sqrt{-1}

2\sqrt{-2}

-2\sqrt{-2}

6 eigenvectors of ad H:

(0,0,0,1,1,0)

(1,1,1,0,0,1)

(−−1,−1,−1,0,0,1)

(−1,−1,−−1,0,0,1)

(−1,1,−1,−−2,−2,1)

(−1,1,−1,−2,−−2,1)

Centralizer type: 2A^{4}_1
Reductive components (2 total):
Scalar product computed:

\begin{pmatrix}1/32\\ \end{pmatrix}

Simple basis of Cartan of centralizer (1 total):

-\sqrt{-1}g_{11}-\sqrt{-1}g_{10}+\sqrt{-1}g_{-10}+\sqrt{-1}g_{-11}

matching e:

g_{16}+\sqrt{-1}g_{5}-g_{4}-\sqrt{-1}g_{3}+\sqrt{-1}g_{-3}+g_{-4}-\sqrt{-1}g_{-5}-g_{-16}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & \sqrt{-1} & 0 & 0 & 0 & -\sqrt{-1}\\ -\sqrt{-1} & 0 & \sqrt{-1} & 0 & 0 & 0\\ 0 & -\sqrt{-1} & 0 & 0 & 0 & \sqrt{-1}\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ \sqrt{-1} & 0 & -\sqrt{-1} & 0 & 0 & 0\\ \end{pmatrix}

Linear space basis of intersection of centralizer and ambient Cartan:

-\sqrt{-1}g_{11}-\sqrt{-1}g_{10}+\sqrt{-1}g_{-10}+\sqrt{-1}g_{-11}

matching e:

g_{16}+\sqrt{-1}g_{5}-g_{4}-\sqrt{-1}g_{3}+\sqrt{-1}g_{-3}+g_{-4}-\sqrt{-1}g_{-5}-g_{-16}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & \sqrt{-1} & 0 & 0 & 0 & -\sqrt{-1}\\ -\sqrt{-1} & 0 & \sqrt{-1} & 0 & 0 & 0\\ 0 & -\sqrt{-1} & 0 & 0 & 0 & \sqrt{-1}\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ \sqrt{-1} & 0 & -\sqrt{-1} & 0 & 0 & 0\\ \end{pmatrix}

Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form:

\begin{pmatrix}128\\ \end{pmatrix}

Scalar product computed:

\begin{pmatrix}1/32\\ \end{pmatrix}

Simple basis of Cartan of centralizer (1 total):

-1/2\sqrt{-2}g_{16}-1/2\sqrt{-2}g_{5}-1/2\sqrt{-2}g_{4}-1/2\sqrt{-2}g_{3}+1/2\sqrt{-2}g_{-3}+1/2\sqrt{-2}g_{-4}+1/2\sqrt{-2}g_{-5}+1/2\sqrt{-2}g_{-16}

matching e:

g_{16}+\sqrt{-2}g_{11}-\sqrt{-2}g_{10}-g_{5}+g_{4}-g_{3}+g_{-3}-g_{-4}+g_{-5}+\sqrt{-2}g_{-10}-\sqrt{-2}g_{-11}-g_{-16}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & -1/2\sqrt{-2} & 1/2\sqrt{-2} & 0\\ 0 & 0 & 0 & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 0\\ 0 & 0 & 0 & -1/2\sqrt{-2} & 1/2\sqrt{-2} & 0\\ 1/2\sqrt{-2} & -1/2\sqrt{-2} & 1/2\sqrt{-2} & 0 & 0 & -1/2\sqrt{-2}\\ -1/2\sqrt{-2} & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 0 & 0 & 1/2\sqrt{-2}\\ 0 & 0 & 0 & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 0\\ \end{pmatrix}

Linear space basis of intersection of centralizer and ambient Cartan:

-1/2\sqrt{-2}g_{16}-1/2\sqrt{-2}g_{5}-1/2\sqrt{-2}g_{4}-1/2\sqrt{-2}g_{3}+1/2\sqrt{-2}g_{-3}+1/2\sqrt{-2}g_{-4}+1/2\sqrt{-2}g_{-5}+1/2\sqrt{-2}g_{-16}

matching e:

g_{16}+\sqrt{-2}g_{11}-\sqrt{-2}g_{10}-g_{5}+g_{4}-g_{3}+g_{-3}-g_{-4}+g_{-5}+\sqrt{-2}g_{-10}-\sqrt{-2}g_{-11}-g_{-16}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & -1/2\sqrt{-2} & 1/2\sqrt{-2} & 0\\ 0 & 0 & 0 & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 0\\ 0 & 0 & 0 & -1/2\sqrt{-2} & 1/2\sqrt{-2} & 0\\ 1/2\sqrt{-2} & -1/2\sqrt{-2} & 1/2\sqrt{-2} & 0 & 0 & -1/2\sqrt{-2}\\ -1/2\sqrt{-2} & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 0 & 0 & 1/2\sqrt{-2}\\ 0 & 0 & 0 & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 0\\ \end{pmatrix}

Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form:

\begin{pmatrix}128\\ \end{pmatrix}

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&13h_{7}+24h_{6}+22h_{5}+20h_{4}+18h_{3}+16h_{2}+10h_{1}\\ e&=&x_{4} g_{43}+x_{5} g_{37}+x_{6} g_{29}+x_{2} g_{26}+x_{7} g_{19}+x_{3} g_{7}+x_{1} g_{1}\\ f&=&x_{8} g_{-1}+x_{10} g_{-7}+x_{14} g_{-19}+x_{9} g_{-26}+x_{13} g_{-29}+x_{12} g_{-37}+x_{11} g_{-43}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -13\right)h_{7}+\left(2x_{2} x_{9} +2x_{4} x_{11} +2x_{5} x_{12} +2x_{6} x_{13} +2x_{7} x_{14} -24\right)h_{6}+\left(2x_{2} x_{9} +2x_{4} x_{11} +2x_{5} x_{12} +2x_{6} x_{13} -22\right)h_{5}+\left(2x_{2} x_{9} +2x_{4} x_{11} +2x_{5} x_{12} -20\right)h_{4}+\left(2x_{2} x_{9} +2x_{4} x_{11} -18\right)h_{3}+\left(2x_{2} x_{9} -16\right)h_{2}+\left(2x_{1} x_{8} -10\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{8} -10&=&0\\2x_{2} x_{9} -16&=&0\\2x_{2} x_{9} +2x_{4} x_{11} -18&=&0\\2x_{2} x_{9} +2x_{4} x_{11} +2x_{5} x_{12} -20&=&0\\2x_{2} x_{9} +2x_{4} x_{11} +2x_{5} x_{12} +2x_{6} x_{13} -22&=&0\\2x_{2} x_{9} +2x_{4} x_{11} +2x_{5} x_{12} +2x_{6} x_{13} +2x_{7} x_{14} -24&=&0\\x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -13&=&0\\\end{array}

\begin{array}{rcl}h&=&13h_{7}+24h_{6}+22h_{5}+20h_{4}+18h_{3}+16h_{2}+10h_{1}\\e&=&x_{4} g_{43}+x_{5} g_{37}+x_{6} g_{29}+x_{2} g_{26}+x_{7} g_{19}+x_{3} g_{7}+x_{1} g_{1}\\f&=&g_{-1}+g_{-7}+g_{-19}+g_{-26}+g_{-29}+g_{-37}+g_{-43}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 2 & 0 & 0 & 0\\ 0 & 2 & 0 & 2 & 2 & 0 & 0\\ 0 & 2 & 0 & 2 & 2 & 2 & 0\\ 0 & 2 & 0 & 2 & 2 & 2 & 2\\ 0 & 0 & 1 & 1 & 1 & 1 & 1\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}10\\ 16\\ 18\\ 20\\ 22\\ 24\\ 13\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&13h_{7}+24h_{6}+22h_{5}+20h_{4}+18h_{3}+16h_{2}+10h_{1}\\ e&=&x_{4} g_{43}+x_{5} g_{37}+x_{6} g_{29}+x_{2} g_{26}+x_{7} g_{19}+x_{3} g_{7}+x_{1} g_{1}\\ f&=&x_{8} g_{-1}+x_{10} g_{-7}+x_{14} g_{-19}+x_{9} g_{-26}+x_{13} g_{-29}+x_{12} g_{-37}+x_{11} g_{-43}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{8} -10&=&0\\2x_{2} x_{9} -16&=&0\\2x_{2} x_{9} +2x_{4} x_{11} -18&=&0\\2x_{2} x_{9} +2x_{4} x_{11} +2x_{5} x_{12} -20&=&0\\2x_{2} x_{9} +2x_{4} x_{11} +2x_{5} x_{12} +2x_{6} x_{13} -22&=&0\\2x_{2} x_{9} +2x_{4} x_{11} +2x_{5} x_{12} +2x_{6} x_{13} +2x_{7} x_{14} -24&=&0\\x_{3} x_{10} +x_{4} x_{11} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -13&=&0\\\end{array}

A^{38}_1

h-characteristic: (2, 2, 0, 0, 0, 1, 0)
Length of the weight dual to h: 76
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 4
Containing regular semisimple subalgebra number 1:

C^{1}_3+3A^{1}_1

Containing regular semisimple subalgebra number 2:

C^{1}_3+A^{2}_1+A^{1}_1

Containing regular semisimple subalgebra number 3:

C^{1}_4+2A^{1}_1

Containing regular semisimple subalgebra number 4:

C^{1}_4+A^{2}_1

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

V_{10\psi}+4V_{6\psi}+2V_{5\psi}+3V_{4\psi}+7V_{2\psi}+6V_{\psi}+6V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&12h_{7}+24h_{6}+22h_{5}+20h_{4}+18h_{3}+16h_{2}+10h_{1}\\ e&=&g_{43}+g_{37}+g_{29}+8g_{21}+9g_{19}+5g_{1}\\ f&=&g_{-1}+g_{-19}+g_{-21}+g_{-29}+g_{-37}+g_{-43}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&12h_{7}+24h_{6}+22h_{5}+20h_{4}+18h_{3}+16h_{2}+10h_{1}\\ [h, e]&=&2g_{43}+2g_{37}+2g_{29}+16g_{21}+18g_{19}+10g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-19}-2g_{-21}-2g_{-29}-2g_{-37}-2g_{-43}\end{array}

Centralizer type:

A^{8}_1+A_1

Killing form square of Cartan element dual to ambient long root: 32
Basis of the centralizer (dimension: 6):

g_{-7}

h_{7}

g_{3}-g_{-3}

g_{4}-g_{-4}

g_{7}

g_{10}-g_{-10}

Basis of centralizer intersected with cartan (dimension: 1):

-h_{7}

Cartan of centralizer (dimension: 2):

-g_{7}-h_{7}-g_{-7}

g_{10}+g_{4}+g_{3}-g_{-3}-g_{-4}-g_{-10}

Cartan-generating semisimple element:

g_{10}+g_{7}+g_{4}+g_{3}+h_{7}-g_{-3}-g_{-4}+g_{-7}-g_{-10}

adjoint action:

\begin{pmatrix}-2 & 2 & 0 & 0 & 0 & 0\\ 1 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & -1 & 0 & 1\\ 0 & 0 & 1 & 0 & 0 & -1\\ 0 & -2 & 0 & 0 & 2 & 0\\ 0 & 0 & -1 & 1 & 0 & 0\\ \end{pmatrix}

Characteristic polynomial ad H:

x^6-5x^4-24x^2

Factorization of characteristic polynomial of ad H: (x )(x )(x^2-8)(x^2+3)
Eigenvalues of ad H:

0

2\sqrt{2}

-2\sqrt{2}

\sqrt{-3}

-\sqrt{-3}

6 eigenvectors of ad H:

(1,1,0,0,1,0)

(0,0,1,1,0,1)

(22−3,−2+1,0,0,1,0)

(−22−3,2+1,0,0,1,0)

(0,0,−12−3−12,12−3−12,0,1)

(0,0,12−3−12,−12−3−12,0,1)

Centralizer type: A^{8}_1+A^{1}_1
Reductive components (2 total):
Scalar product computed:

\begin{pmatrix}1/8\\ \end{pmatrix}

Simple basis of Cartan of centralizer (1 total):

1/2\sqrt{2}g_{7}+1/2\sqrt{2}h_{7}+1/2\sqrt{2}g_{-7}

matching e:

g_{7}+\left(-\sqrt{2}+1\right)h_{7}+\left(2\sqrt{2}-3\right)g_{-7}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}-\sqrt{2} & \sqrt{2} & 0 & 0 & 0 & 0\\ 1/2\sqrt{2} & 0 & 0 & 0 & -1/2\sqrt{2} & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -\sqrt{2} & 0 & 0 & \sqrt{2} & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

Linear space basis of intersection of centralizer and ambient Cartan:

1/2\sqrt{2}g_{7}+1/2\sqrt{2}h_{7}+1/2\sqrt{2}g_{-7}

matching e:

g_{7}+\left(-\sqrt{2}+1\right)h_{7}+\left(2\sqrt{2}-3\right)g_{-7}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}-\sqrt{2} & \sqrt{2} & 0 & 0 & 0 & 0\\ 1/2\sqrt{2} & 0 & 0 & 0 & -1/2\sqrt{2} & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -\sqrt{2} & 0 & 0 & \sqrt{2} & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form:

\begin{pmatrix}32\\ \end{pmatrix}

Scalar product computed:

\begin{pmatrix}1/64\\ \end{pmatrix}

Simple basis of Cartan of centralizer (1 total):

-2/3\sqrt{-3}g_{10}-2/3\sqrt{-3}g_{4}-2/3\sqrt{-3}g_{3}+2/3\sqrt{-3}g_{-3}+2/3\sqrt{-3}g_{-4}+2/3\sqrt{-3}g_{-10}

matching e:

g_{10}+\left(1/2\sqrt{-3}-1/2\right)g_{4}+\left(-1/2\sqrt{-3}-1/2\right)g_{3}+\left(1/2\sqrt{-3}+1/2\right)g_{-3}+\left(-1/2\sqrt{-3}+1/2\right)g_{-4}-g_{-10}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2/3\sqrt{-3} & 0 & -2/3\sqrt{-3}\\ 0 & 0 & -2/3\sqrt{-3} & 0 & 0 & 2/3\sqrt{-3}\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2/3\sqrt{-3} & -2/3\sqrt{-3} & 0 & 0\\ \end{pmatrix}

Linear space basis of intersection of centralizer and ambient Cartan:

-2/3\sqrt{-3}g_{10}-2/3\sqrt{-3}g_{4}-2/3\sqrt{-3}g_{3}+2/3\sqrt{-3}g_{-3}+2/3\sqrt{-3}g_{-4}+2/3\sqrt{-3}g_{-10}

matching e:

g_{10}+\left(1/2\sqrt{-3}-1/2\right)g_{4}+\left(-1/2\sqrt{-3}-1/2\right)g_{3}+\left(1/2\sqrt{-3}+1/2\right)g_{-3}+\left(-1/2\sqrt{-3}+1/2\right)g_{-4}-g_{-10}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2/3\sqrt{-3} & 0 & -2/3\sqrt{-3}\\ 0 & 0 & -2/3\sqrt{-3} & 0 & 0 & 2/3\sqrt{-3}\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2/3\sqrt{-3} & -2/3\sqrt{-3} & 0 & 0\\ \end{pmatrix}

Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form:

\begin{pmatrix}256\\ \end{pmatrix}

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&12h_{7}+24h_{6}+22h_{5}+20h_{4}+18h_{3}+16h_{2}+10h_{1}\\ e&=&x_{4} g_{43}+x_{5} g_{37}+x_{6} g_{29}+x_{2} g_{21}+x_{3} g_{19}+x_{1} g_{1}\\ f&=&x_{7} g_{-1}+x_{9} g_{-19}+x_{8} g_{-21}+x_{12} g_{-29}+x_{11} g_{-37}+x_{10} g_{-43}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -12\right)h_{7}+\left(2x_{3} x_{9} +2x_{4} x_{10} +2x_{5} x_{11} +2x_{6} x_{12} -24\right)h_{6}+\left(2x_{2} x_{8} +2x_{4} x_{10} +2x_{5} x_{11} +2x_{6} x_{12} -22\right)h_{5}+\left(2x_{2} x_{8} +2x_{4} x_{10} +2x_{5} x_{11} -20\right)h_{4}+\left(2x_{2} x_{8} +2x_{4} x_{10} -18\right)h_{3}+\left(2x_{2} x_{8} -16\right)h_{2}+\left(2x_{1} x_{7} -10\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{7} -10&=&0\\2x_{2} x_{8} -16&=&0\\2x_{2} x_{8} +2x_{4} x_{10} -18&=&0\\2x_{2} x_{8} +2x_{4} x_{10} +2x_{5} x_{11} -20&=&0\\2x_{2} x_{8} +2x_{4} x_{10} +2x_{5} x_{11} +2x_{6} x_{12} -22&=&0\\2x_{3} x_{9} +2x_{4} x_{10} +2x_{5} x_{11} +2x_{6} x_{12} -24&=&0\\x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -12&=&0\\\end{array}

\begin{array}{rcl}h&=&12h_{7}+24h_{6}+22h_{5}+20h_{4}+18h_{3}+16h_{2}+10h_{1}\\e&=&x_{4} g_{43}+x_{5} g_{37}+x_{6} g_{29}+x_{2} g_{21}+x_{3} g_{19}+x_{1} g_{1}\\f&=&g_{-1}+g_{-19}+g_{-21}+g_{-29}+g_{-37}+g_{-43}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 2 & 0 & 0\\ 0 & 2 & 0 & 2 & 2 & 0\\ 0 & 2 & 0 & 2 & 2 & 2\\ 0 & 0 & 2 & 2 & 2 & 2\\ 0 & 0 & 1 & 1 & 1 & 1\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}10\\ 16\\ 18\\ 20\\ 22\\ 24\\ 12\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&12h_{7}+24h_{6}+22h_{5}+20h_{4}+18h_{3}+16h_{2}+10h_{1}\\ e&=&x_{4} g_{43}+x_{5} g_{37}+x_{6} g_{29}+x_{2} g_{21}+x_{3} g_{19}+x_{1} g_{1}\\ f&=&x_{7} g_{-1}+x_{9} g_{-19}+x_{8} g_{-21}+x_{12} g_{-29}+x_{11} g_{-37}+x_{10} g_{-43}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{7} -10&=&0\\2x_{2} x_{8} -16&=&0\\2x_{2} x_{8} +2x_{4} x_{10} -18&=&0\\2x_{2} x_{8} +2x_{4} x_{10} +2x_{5} x_{11} -20&=&0\\2x_{2} x_{8} +2x_{4} x_{10} +2x_{5} x_{11} +2x_{6} x_{12} -22&=&0\\2x_{3} x_{9} +2x_{4} x_{10} +2x_{5} x_{11} +2x_{6} x_{12} -24&=&0\\x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -12&=&0\\\end{array}

A^{37}_1

h-characteristic: (2, 2, 0, 0, 1, 0, 0)
Length of the weight dual to h: 74
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1:

C^{1}_3+2A^{1}_1

Containing regular semisimple subalgebra number 2:

C^{1}_3+A^{2}_1

Containing regular semisimple subalgebra number 3:

C^{1}_4+A^{1}_1

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

V_{10\psi}+3V_{6\psi}+4V_{5\psi}+2V_{4\psi}+4V_{2\psi}+8V_{\psi}+11V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&11h_{7}+22h_{6}+22h_{5}+20h_{4}+18h_{3}+16h_{2}+10h_{1}\\ e&=&g_{43}+g_{37}+9g_{29}+8g_{15}+5g_{1}\\ f&=&g_{-1}+g_{-15}+g_{-29}+g_{-37}+g_{-43}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&11h_{7}+22h_{6}+22h_{5}+20h_{4}+18h_{3}+16h_{2}+10h_{1}\\ [h, e]&=&2g_{43}+2g_{37}+18g_{29}+16g_{15}+10g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-15}-2g_{-29}-2g_{-37}-2g_{-43}\end{array}

Centralizer type:

B_2

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&11h_{7}+22h_{6}+22h_{5}+20h_{4}+18h_{3}+16h_{2}+10h_{1}\\ e&=&x_{4} g_{43}+x_{5} g_{37}+x_{3} g_{29}+x_{2} g_{15}+x_{1} g_{1}\\ f&=&x_{6} g_{-1}+x_{7} g_{-15}+x_{8} g_{-29}+x_{10} g_{-37}+x_{9} g_{-43}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(x_{3} x_{8} +x_{4} x_{9} +x_{5} x_{10} -11\right)h_{7}+\left(2x_{3} x_{8} +2x_{4} x_{9} +2x_{5} x_{10} -22\right)h_{6}+\left(2x_{3} x_{8} +2x_{4} x_{9} +2x_{5} x_{10} -22\right)h_{5}+\left(2x_{2} x_{7} +2x_{4} x_{9} +2x_{5} x_{10} -20\right)h_{4}+\left(2x_{2} x_{7} +2x_{4} x_{9} -18\right)h_{3}+\left(2x_{2} x_{7} -16\right)h_{2}+\left(2x_{1} x_{6} -10\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{6} -10&=&0\\2x_{2} x_{7} -16&=&0\\2x_{2} x_{7} +2x_{4} x_{9} -18&=&0\\2x_{2} x_{7} +2x_{4} x_{9} +2x_{5} x_{10} -20&=&0\\2x_{3} x_{8} +2x_{4} x_{9} +2x_{5} x_{10} -22&=&0\\2x_{3} x_{8} +2x_{4} x_{9} +2x_{5} x_{10} -22&=&0\\x_{3} x_{8} +x_{4} x_{9} +x_{5} x_{10} -11&=&0\\\end{array}

\begin{array}{rcl}h&=&11h_{7}+22h_{6}+22h_{5}+20h_{4}+18h_{3}+16h_{2}+10h_{1}\\e&=&x_{4} g_{43}+x_{5} g_{37}+x_{3} g_{29}+x_{2} g_{15}+x_{1} g_{1}\\f&=&g_{-1}+g_{-15}+g_{-29}+g_{-37}+g_{-43}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0\\ 0 & 2 & 0 & 2 & 0\\ 0 & 2 & 0 & 2 & 2\\ 0 & 0 & 2 & 2 & 2\\ 0 & 0 & 2 & 2 & 2\\ 0 & 0 & 1 & 1 & 1\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}10\\ 16\\ 18\\ 20\\ 22\\ 22\\ 11\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&11h_{7}+22h_{6}+22h_{5}+20h_{4}+18h_{3}+16h_{2}+10h_{1}\\ e&=&x_{4} g_{43}+x_{5} g_{37}+x_{3} g_{29}+x_{2} g_{15}+x_{1} g_{1}\\ f&=&x_{6} g_{-1}+x_{7} g_{-15}+x_{8} g_{-29}+x_{10} g_{-37}+x_{9} g_{-43}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{6} -10&=&0\\2x_{2} x_{7} -16&=&0\\2x_{2} x_{7} +2x_{4} x_{9} -18&=&0\\2x_{2} x_{7} +2x_{4} x_{9} +2x_{5} x_{10} -20&=&0\\2x_{3} x_{8} +2x_{4} x_{9} +2x_{5} x_{10} -22&=&0\\2x_{3} x_{8} +2x_{4} x_{9} +2x_{5} x_{10} -22&=&0\\x_{3} x_{8} +x_{4} x_{9} +x_{5} x_{10} -11&=&0\\\end{array}

A^{36}_1

h-characteristic: (2, 2, 0, 1, 0, 0, 0)
Length of the weight dual to h: 72
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1:

C^{1}_3+A^{1}_1

Containing regular semisimple subalgebra number 2:

C^{1}_4

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

V_{10\psi}+2V_{6\psi}+6V_{5\psi}+V_{4\psi}+2V_{2\psi}+6V_{\psi}+21V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&10h_{7}+20h_{6}+20h_{5}+20h_{4}+18h_{3}+16h_{2}+10h_{1}\\ e&=&g_{43}+9g_{37}+8g_{9}+5g_{1}\\ f&=&g_{-1}+g_{-9}+g_{-37}+g_{-43}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&10h_{7}+20h_{6}+20h_{5}+20h_{4}+18h_{3}+16h_{2}+10h_{1}\\ [h, e]&=&2g_{43}+18g_{37}+16g_{9}+10g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-9}-2g_{-37}-2g_{-43}\end{array}

Centralizer type:

C_3

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&10h_{7}+20h_{6}+20h_{5}+20h_{4}+18h_{3}+16h_{2}+10h_{1}\\ e&=&x_{4} g_{43}+x_{3} g_{37}+x_{2} g_{9}+x_{1} g_{1}\\ f&=&x_{5} g_{-1}+x_{6} g_{-9}+x_{7} g_{-37}+x_{8} g_{-43}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(x_{3} x_{7} +x_{4} x_{8} -10\right)h_{7}+\left(2x_{3} x_{7} +2x_{4} x_{8} -20\right)h_{6}+\left(2x_{3} x_{7} +2x_{4} x_{8} -20\right)h_{5}+\left(2x_{3} x_{7} +2x_{4} x_{8} -20\right)h_{4}+\left(2x_{2} x_{6} +2x_{4} x_{8} -18\right)h_{3}+\left(2x_{2} x_{6} -16\right)h_{2}+\left(2x_{1} x_{5} -10\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{5} -10&=&0\\2x_{2} x_{6} -16&=&0\\2x_{2} x_{6} +2x_{4} x_{8} -18&=&0\\2x_{3} x_{7} +2x_{4} x_{8} -20&=&0\\2x_{3} x_{7} +2x_{4} x_{8} -20&=&0\\2x_{3} x_{7} +2x_{4} x_{8} -20&=&0\\x_{3} x_{7} +x_{4} x_{8} -10&=&0\\\end{array}

\begin{array}{rcl}h&=&10h_{7}+20h_{6}+20h_{5}+20h_{4}+18h_{3}+16h_{2}+10h_{1}\\e&=&x_{4} g_{43}+x_{3} g_{37}+x_{2} g_{9}+x_{1} g_{1}\\f&=&g_{-1}+g_{-9}+g_{-37}+g_{-43}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0 & 0\\ 0 & 2 & 0 & 0\\ 0 & 2 & 0 & 2\\ 0 & 0 & 2 & 2\\ 0 & 0 & 2 & 2\\ 0 & 0 & 2 & 2\\ 0 & 0 & 1 & 1\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}10\\ 16\\ 18\\ 20\\ 20\\ 20\\ 10\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&10h_{7}+20h_{6}+20h_{5}+20h_{4}+18h_{3}+16h_{2}+10h_{1}\\ e&=&x_{4} g_{43}+x_{3} g_{37}+x_{2} g_{9}+x_{1} g_{1}\\ f&=&x_{5} g_{-1}+x_{6} g_{-9}+x_{7} g_{-37}+x_{8} g_{-43}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{5} -10&=&0\\2x_{2} x_{6} -16&=&0\\2x_{2} x_{6} +2x_{4} x_{8} -18&=&0\\2x_{3} x_{7} +2x_{4} x_{8} -20&=&0\\2x_{3} x_{7} +2x_{4} x_{8} -20&=&0\\2x_{3} x_{7} +2x_{4} x_{8} -20&=&0\\x_{3} x_{7} +x_{4} x_{8} -10&=&0\\\end{array}

A^{35}_1

h-characteristic: (2, 2, 1, 0, 0, 0, 0)
Length of the weight dual to h: 70
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1:

C^{1}_3

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

V_{10\psi}+V_{6\psi}+8V_{5\psi}+V_{2\psi}+36V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&9h_{7}+18h_{6}+18h_{5}+18h_{4}+18h_{3}+16h_{2}+10h_{1}\\ e&=&9g_{43}+8g_{2}+5g_{1}\\ f&=&g_{-1}+g_{-2}+g_{-43}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&9h_{7}+18h_{6}+18h_{5}+18h_{4}+18h_{3}+16h_{2}+10h_{1}\\ [h, e]&=&18g_{43}+16g_{2}+10g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-2}-2g_{-43}\end{array}

Centralizer type:

C_4

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&9h_{7}+18h_{6}+18h_{5}+18h_{4}+18h_{3}+16h_{2}+10h_{1}\\ e&=&x_{3} g_{43}+x_{2} g_{2}+x_{1} g_{1}\\ f&=&x_{4} g_{-1}+x_{5} g_{-2}+x_{6} g_{-43}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(x_{3} x_{6} -9\right)h_{7}+\left(2x_{3} x_{6} -18\right)h_{6}+\left(2x_{3} x_{6} -18\right)h_{5}+\left(2x_{3} x_{6} -18\right)h_{4}+\left(2x_{3} x_{6} -18\right)h_{3}+\left(2x_{2} x_{5} -16\right)h_{2}+\left(2x_{1} x_{4} -10\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{4} -10&=&0\\2x_{2} x_{5} -16&=&0\\2x_{3} x_{6} -18&=&0\\2x_{3} x_{6} -18&=&0\\2x_{3} x_{6} -18&=&0\\2x_{3} x_{6} -18&=&0\\x_{3} x_{6} -9&=&0\\\end{array}

\begin{array}{rcl}h&=&9h_{7}+18h_{6}+18h_{5}+18h_{4}+18h_{3}+16h_{2}+10h_{1}\\e&=&x_{3} g_{43}+x_{2} g_{2}+x_{1} g_{1}\\f&=&g_{-1}+g_{-2}+g_{-43}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0\\ 0 & 2 & 0\\ 0 & 0 & 2\\ 0 & 0 & 2\\ 0 & 0 & 2\\ 0 & 0 & 2\\ 0 & 0 & 1\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}10\\ 16\\ 18\\ 18\\ 18\\ 18\\ 9\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&9h_{7}+18h_{6}+18h_{5}+18h_{4}+18h_{3}+16h_{2}+10h_{1}\\ e&=&x_{3} g_{43}+x_{2} g_{2}+x_{1} g_{1}\\ f&=&x_{4} g_{-1}+x_{5} g_{-2}+x_{6} g_{-43}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{4} -10&=&0\\2x_{2} x_{5} -16&=&0\\2x_{3} x_{6} -18&=&0\\2x_{3} x_{6} -18&=&0\\2x_{3} x_{6} -18&=&0\\2x_{3} x_{6} -18&=&0\\x_{3} x_{6} -9&=&0\\\end{array}

A^{31}_1

h-characteristic: (0, 0, 2, 0, 0, 0, 2)
Length of the weight dual to h: 62
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 4
Containing regular semisimple subalgebra number 1:

3B^{1}_2+A^{1}_1

Containing regular semisimple subalgebra number 2:

A^{2}_3+B^{1}_2+A^{1}_1

Containing regular semisimple subalgebra number 3:

C^{1}_3+2B^{1}_2

Containing regular semisimple subalgebra number 4:

A^{2}_3+C^{1}_3

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

6V_{6\psi}+6V_{4\psi}+10V_{2\psi}+3V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&13h_{7}+24h_{6}+22h_{5}+20h_{4}+18h_{3}+12h_{2}+6h_{1}\\ e&=&4g_{37}+4g_{29}+4g_{19}+3g_{16}+3g_{15}+3g_{14}+g_{7}\\ f&=&g_{-7}+g_{-14}+g_{-15}+g_{-16}+g_{-19}+g_{-29}+g_{-37}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&13h_{7}+24h_{6}+22h_{5}+20h_{4}+18h_{3}+12h_{2}+6h_{1}\\ [h, e]&=&8g_{37}+8g_{29}+8g_{19}+6g_{16}+6g_{15}+6g_{14}+2g_{7}\\ [h, f]&=&-2g_{-7}-2g_{-14}-2g_{-15}-2g_{-16}-2g_{-19}-2g_{-29}-2g_{-37}\end{array}

Centralizer type:

A^{16}_1

Killing form square of Cartan element dual to ambient long root: 32
Basis of the centralizer (dimension: 3):

g_{4}+g_{1}-g_{-1}-g_{-4}

g_{5}+g_{2}-g_{-2}-g_{-5}

g_{11}+g_{8}-g_{-8}-g_{-11}

Basis of centralizer intersected with cartan (dimension: 0):
Cartan of centralizer (dimension: 1):

g_{11}+g_{8}+g_{5}+g_{4}+g_{2}+g_{1}-g_{-1}-g_{-2}-g_{-4}-g_{-5}-g_{-8}-g_{-11}

Cartan-generating semisimple element:

g_{11}+g_{8}+g_{5}+g_{4}+g_{2}+g_{1}-g_{-1}-g_{-2}-g_{-4}-g_{-5}-g_{-8}-g_{-11}

adjoint action:

\begin{pmatrix}0 & -1 & 1\\ 1 & 0 & -1\\ -1 & 1 & 0\\ \end{pmatrix}

Characteristic polynomial ad H:

x^3+3x

Factorization of characteristic polynomial of ad H: (x )(x^2+3)
Eigenvalues of ad H:

0

\sqrt{-3}

-\sqrt{-3}

3 eigenvectors of ad H:

(1,1,1)

(−12−3−12,12−3−12,1)

(12−3−12,−12−3−12,1)

Centralizer type: A^{16}_1
Reductive components (1 total):
Scalar product computed:

\begin{pmatrix}1/128\\ \end{pmatrix}

Simple basis of Cartan of centralizer (1 total):

-2/3\sqrt{-3}g_{11}-2/3\sqrt{-3}g_{8}-2/3\sqrt{-3}g_{5}-2/3\sqrt{-3}g_{4}-2/3\sqrt{-3}g_{2}-2/3\sqrt{-3}g_{1}+2/3\sqrt{-3}g_{-1}+2/3\sqrt{-3}g_{-2}+2/3\sqrt{-3}g_{-4}+2/3\sqrt{-3}g_{-5}+2/3\sqrt{-3}g_{-8}+2/3\sqrt{-3}g_{-11}

matching e:

g_{11}+g_{8}+\left(1/2\sqrt{-3}-1/2\right)g_{5}+\left(-1/2\sqrt{-3}-1/2\right)g_{4}+\left(1/2\sqrt{-3}-1/2\right)g_{2}+\left(-1/2\sqrt{-3}-1/2\right)g_{1}+\left(1/2\sqrt{-3}+1/2\right)g_{-1}+\left(-1/2\sqrt{-3}+1/2\right)g_{-2}+\left(1/2\sqrt{-3}+1/2\right)g_{-4}+\left(-1/2\sqrt{-3}+1/2\right)g_{-5}-g_{-8}-g_{-11}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 2/3\sqrt{-3} & -2/3\sqrt{-3}\\ -2/3\sqrt{-3} & 0 & 2/3\sqrt{-3}\\ 2/3\sqrt{-3} & -2/3\sqrt{-3} & 0\\ \end{pmatrix}

Linear space basis of intersection of centralizer and ambient Cartan:

-2/3\sqrt{-3}g_{11}-2/3\sqrt{-3}g_{8}-2/3\sqrt{-3}g_{5}-2/3\sqrt{-3}g_{4}-2/3\sqrt{-3}g_{2}-2/3\sqrt{-3}g_{1}+2/3\sqrt{-3}g_{-1}+2/3\sqrt{-3}g_{-2}+2/3\sqrt{-3}g_{-4}+2/3\sqrt{-3}g_{-5}+2/3\sqrt{-3}g_{-8}+2/3\sqrt{-3}g_{-11}

matching e:

g_{11}+g_{8}+\left(1/2\sqrt{-3}-1/2\right)g_{5}+\left(-1/2\sqrt{-3}-1/2\right)g_{4}+\left(1/2\sqrt{-3}-1/2\right)g_{2}+\left(-1/2\sqrt{-3}-1/2\right)g_{1}+\left(1/2\sqrt{-3}+1/2\right)g_{-1}+\left(-1/2\sqrt{-3}+1/2\right)g_{-2}+\left(1/2\sqrt{-3}+1/2\right)g_{-4}+\left(-1/2\sqrt{-3}+1/2\right)g_{-5}-g_{-8}-g_{-11}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 2/3\sqrt{-3} & -2/3\sqrt{-3}\\ -2/3\sqrt{-3} & 0 & 2/3\sqrt{-3}\\ 2/3\sqrt{-3} & -2/3\sqrt{-3} & 0\\ \end{pmatrix}

Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form:

\begin{pmatrix}512\\ \end{pmatrix}

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&13h_{7}+24h_{6}+22h_{5}+20h_{4}+18h_{3}+12h_{2}+6h_{1}\\ e&=&x_{1} g_{37}+x_{3} g_{29}+x_{5} g_{19}+x_{6} g_{16}+x_{4} g_{15}+x_{2} g_{14}+x_{7} g_{7}\\ f&=&x_{14} g_{-7}+x_{9} g_{-14}+x_{11} g_{-15}+x_{13} g_{-16}+x_{12} g_{-19}+x_{10} g_{-29}+x_{8} g_{-37}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(x_{1} x_{8} +x_{3} x_{10} +x_{5} x_{12} +x_{7} x_{14} -13\right)h_{7}+\left(2x_{1} x_{8} +2x_{3} x_{10} +2x_{5} x_{12} -24\right)h_{6}+\left(2x_{1} x_{8} +2x_{3} x_{10} +2x_{6} x_{13} -22\right)h_{5}+\left(2x_{1} x_{8} +2x_{4} x_{11} +2x_{6} x_{13} -20\right)h_{4}+\left(2x_{2} x_{9} +2x_{4} x_{11} +2x_{6} x_{13} -18\right)h_{3}+\left(2x_{2} x_{9} +2x_{4} x_{11} -12\right)h_{2}+\left(2x_{2} x_{9} -6\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{8} +2x_{4} x_{11} +2x_{6} x_{13} -20&=&0\\2x_{1} x_{8} +2x_{3} x_{10} +2x_{6} x_{13} -22&=&0\\2x_{1} x_{8} +2x_{3} x_{10} +2x_{5} x_{12} -24&=&0\\x_{1} x_{8} +x_{3} x_{10} +x_{5} x_{12} +x_{7} x_{14} -13&=&0\\2x_{2} x_{9} -6&=&0\\2x_{2} x_{9} +2x_{4} x_{11} -12&=&0\\2x_{2} x_{9} +2x_{4} x_{11} +2x_{6} x_{13} -18&=&0\\\end{array}

\begin{array}{rcl}h&=&13h_{7}+24h_{6}+22h_{5}+20h_{4}+18h_{3}+12h_{2}+6h_{1}\\e&=&x_{1} g_{37}+x_{3} g_{29}+x_{5} g_{19}+x_{6} g_{16}+x_{4} g_{15}+x_{2} g_{14}+x_{7} g_{7}\\f&=&g_{-7}+g_{-14}+g_{-15}+g_{-16}+g_{-19}+g_{-29}+g_{-37}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0 & 2 & 0 & 2 & 0\\ 2 & 0 & 2 & 0 & 0 & 2 & 0\\ 2 & 0 & 2 & 0 & 2 & 0 & 0\\ 1 & 0 & 1 & 0 & 1 & 0 & 1\\ 0 & 2 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 2 & 0 & 0 & 0\\ 0 & 2 & 0 & 2 & 0 & 2 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}20\\ 22\\ 24\\ 13\\ 6\\ 12\\ 18\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&13h_{7}+24h_{6}+22h_{5}+20h_{4}+18h_{3}+12h_{2}+6h_{1}\\ e&=&x_{1} g_{37}+x_{3} g_{29}+x_{5} g_{19}+x_{6} g_{16}+x_{4} g_{15}+x_{2} g_{14}+x_{7} g_{7}\\ f&=&x_{14} g_{-7}+x_{9} g_{-14}+x_{11} g_{-15}+x_{13} g_{-16}+x_{12} g_{-19}+x_{10} g_{-29}+x_{8} g_{-37}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{8} +2x_{4} x_{11} +2x_{6} x_{13} -20&=&0\\2x_{1} x_{8} +2x_{3} x_{10} +2x_{6} x_{13} -22&=&0\\2x_{1} x_{8} +2x_{3} x_{10} +2x_{5} x_{12} -24&=&0\\x_{1} x_{8} +x_{3} x_{10} +x_{5} x_{12} +x_{7} x_{14} -13&=&0\\2x_{2} x_{9} -6&=&0\\2x_{2} x_{9} +2x_{4} x_{11} -12&=&0\\2x_{2} x_{9} +2x_{4} x_{11} +2x_{6} x_{13} -18&=&0\\\end{array}

A^{30}_1

h-characteristic: (0, 0, 2, 0, 0, 1, 0)
Length of the weight dual to h: 60
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1:

3B^{1}_2

Containing regular semisimple subalgebra number 2:

A^{2}_3+B^{1}_2

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

6V_{6\psi}+3V_{4\psi}+6V_{3\psi}+6V_{2\psi}+6V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&12h_{7}+24h_{6}+22h_{5}+20h_{4}+18h_{3}+12h_{2}+6h_{1}\\ e&=&4g_{37}+4g_{29}+4g_{19}+3g_{16}+3g_{15}+3g_{14}\\ f&=&g_{-14}+g_{-15}+g_{-16}+g_{-19}+g_{-29}+g_{-37}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&12h_{7}+24h_{6}+22h_{5}+20h_{4}+18h_{3}+12h_{2}+6h_{1}\\ [h, e]&=&8g_{37}+8g_{29}+8g_{19}+6g_{16}+6g_{15}+6g_{14}\\ [h, f]&=&-2g_{-14}-2g_{-15}-2g_{-16}-2g_{-19}-2g_{-29}-2g_{-37}\end{array}

Centralizer type:

A^{16}_1+A_1

Killing form square of Cartan element dual to ambient long root: 32
Basis of the centralizer (dimension: 6):

g_{-7}

h_{7}

g_{4}+g_{1}-g_{-1}-g_{-4}

g_{5}+g_{2}-g_{-2}-g_{-5}

g_{7}

g_{11}+g_{8}-g_{-8}-g_{-11}

Basis of centralizer intersected with cartan (dimension: 1):

-h_{7}

Cartan of centralizer (dimension: 2):

-g_{7}-h_{7}-g_{-7}

g_{11}+g_{8}+g_{5}+g_{4}+g_{2}+g_{1}-g_{-1}-g_{-2}-g_{-4}-g_{-5}-g_{-8}-g_{-11}

Cartan-generating semisimple element:

g_{11}+g_{8}+g_{7}+g_{5}+g_{4}+g_{2}+g_{1}+h_{7}-g_{-1}-g_{-2}-g_{-4}-g_{-5}+g_{-7}-g_{-8}-g_{-11}

adjoint action:

\begin{pmatrix}-2 & 2 & 0 & 0 & 0 & 0\\ 1 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & -1 & 0 & 1\\ 0 & 0 & 1 & 0 & 0 & -1\\ 0 & -2 & 0 & 0 & 2 & 0\\ 0 & 0 & -1 & 1 & 0 & 0\\ \end{pmatrix}

Characteristic polynomial ad H:

x^6-5x^4-24x^2

Factorization of characteristic polynomial of ad H: (x )(x )(x^2-8)(x^2+3)
Eigenvalues of ad H:

0

2\sqrt{2}

-2\sqrt{2}

\sqrt{-3}

-\sqrt{-3}

6 eigenvectors of ad H:

(1,1,0,0,1,0)

(0,0,1,1,0,1)

(22−3,−2+1,0,0,1,0)

(−22−3,2+1,0,0,1,0)

(0,0,−12−3−12,12−3−12,0,1)

(0,0,12−3−12,−12−3−12,0,1)

Centralizer type: A^{16}_1+A^{1}_1
Reductive components (2 total):
Scalar product computed:

\begin{pmatrix}1/8\\ \end{pmatrix}

Simple basis of Cartan of centralizer (1 total):

1/2\sqrt{2}g_{7}+1/2\sqrt{2}h_{7}+1/2\sqrt{2}g_{-7}

matching e:

g_{7}+\left(-\sqrt{2}+1\right)h_{7}+\left(2\sqrt{2}-3\right)g_{-7}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}-\sqrt{2} & \sqrt{2} & 0 & 0 & 0 & 0\\ 1/2\sqrt{2} & 0 & 0 & 0 & -1/2\sqrt{2} & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -\sqrt{2} & 0 & 0 & \sqrt{2} & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

Linear space basis of intersection of centralizer and ambient Cartan:

1/2\sqrt{2}g_{7}+1/2\sqrt{2}h_{7}+1/2\sqrt{2}g_{-7}

matching e:

g_{7}+\left(-\sqrt{2}+1\right)h_{7}+\left(2\sqrt{2}-3\right)g_{-7}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}-\sqrt{2} & \sqrt{2} & 0 & 0 & 0 & 0\\ 1/2\sqrt{2} & 0 & 0 & 0 & -1/2\sqrt{2} & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -\sqrt{2} & 0 & 0 & \sqrt{2} & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form:

\begin{pmatrix}32\\ \end{pmatrix}

Scalar product computed:

\begin{pmatrix}1/128\\ \end{pmatrix}

Simple basis of Cartan of centralizer (1 total):

-2/3\sqrt{-3}g_{11}-2/3\sqrt{-3}g_{8}-2/3\sqrt{-3}g_{5}-2/3\sqrt{-3}g_{4}-2/3\sqrt{-3}g_{2}-2/3\sqrt{-3}g_{1}+2/3\sqrt{-3}g_{-1}+2/3\sqrt{-3}g_{-2}+2/3\sqrt{-3}g_{-4}+2/3\sqrt{-3}g_{-5}+2/3\sqrt{-3}g_{-8}+2/3\sqrt{-3}g_{-11}

matching e:

g_{11}+g_{8}+\left(1/2\sqrt{-3}-1/2\right)g_{5}+\left(-1/2\sqrt{-3}-1/2\right)g_{4}+\left(1/2\sqrt{-3}-1/2\right)g_{2}+\left(-1/2\sqrt{-3}-1/2\right)g_{1}+\left(1/2\sqrt{-3}+1/2\right)g_{-1}+\left(-1/2\sqrt{-3}+1/2\right)g_{-2}+\left(1/2\sqrt{-3}+1/2\right)g_{-4}+\left(-1/2\sqrt{-3}+1/2\right)g_{-5}-g_{-8}-g_{-11}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2/3\sqrt{-3} & 0 & -2/3\sqrt{-3}\\ 0 & 0 & -2/3\sqrt{-3} & 0 & 0 & 2/3\sqrt{-3}\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2/3\sqrt{-3} & -2/3\sqrt{-3} & 0 & 0\\ \end{pmatrix}

Linear space basis of intersection of centralizer and ambient Cartan:

-2/3\sqrt{-3}g_{11}-2/3\sqrt{-3}g_{8}-2/3\sqrt{-3}g_{5}-2/3\sqrt{-3}g_{4}-2/3\sqrt{-3}g_{2}-2/3\sqrt{-3}g_{1}+2/3\sqrt{-3}g_{-1}+2/3\sqrt{-3}g_{-2}+2/3\sqrt{-3}g_{-4}+2/3\sqrt{-3}g_{-5}+2/3\sqrt{-3}g_{-8}+2/3\sqrt{-3}g_{-11}

matching e:

g_{11}+g_{8}+\left(1/2\sqrt{-3}-1/2\right)g_{5}+\left(-1/2\sqrt{-3}-1/2\right)g_{4}+\left(1/2\sqrt{-3}-1/2\right)g_{2}+\left(-1/2\sqrt{-3}-1/2\right)g_{1}+\left(1/2\sqrt{-3}+1/2\right)g_{-1}+\left(-1/2\sqrt{-3}+1/2\right)g_{-2}+\left(1/2\sqrt{-3}+1/2\right)g_{-4}+\left(-1/2\sqrt{-3}+1/2\right)g_{-5}-g_{-8}-g_{-11}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2/3\sqrt{-3} & 0 & -2/3\sqrt{-3}\\ 0 & 0 & -2/3\sqrt{-3} & 0 & 0 & 2/3\sqrt{-3}\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2/3\sqrt{-3} & -2/3\sqrt{-3} & 0 & 0\\ \end{pmatrix}

Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form:

\begin{pmatrix}512\\ \end{pmatrix}

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&12h_{7}+24h_{6}+22h_{5}+20h_{4}+18h_{3}+12h_{2}+6h_{1}\\ e&=&x_{1} g_{37}+x_{3} g_{29}+x_{5} g_{19}+x_{6} g_{16}+x_{4} g_{15}+x_{2} g_{14}\\ f&=&x_{8} g_{-14}+x_{10} g_{-15}+x_{12} g_{-16}+x_{11} g_{-19}+x_{9} g_{-29}+x_{7} g_{-37}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(x_{1} x_{7} +x_{3} x_{9} +x_{5} x_{11} -12\right)h_{7}+\left(2x_{1} x_{7} +2x_{3} x_{9} +2x_{5} x_{11} -24\right)h_{6}+\left(2x_{1} x_{7} +2x_{3} x_{9} +2x_{6} x_{12} -22\right)h_{5}+\left(2x_{1} x_{7} +2x_{4} x_{10} +2x_{6} x_{12} -20\right)h_{4}+\left(2x_{2} x_{8} +2x_{4} x_{10} +2x_{6} x_{12} -18\right)h_{3}+\left(2x_{2} x_{8} +2x_{4} x_{10} -12\right)h_{2}+\left(2x_{2} x_{8} -6\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{7} +2x_{4} x_{10} +2x_{6} x_{12} -20&=&0\\2x_{1} x_{7} +2x_{3} x_{9} +2x_{6} x_{12} -22&=&0\\2x_{1} x_{7} +2x_{3} x_{9} +2x_{5} x_{11} -24&=&0\\x_{1} x_{7} +x_{3} x_{9} +x_{5} x_{11} -12&=&0\\2x_{2} x_{8} -6&=&0\\2x_{2} x_{8} +2x_{4} x_{10} -12&=&0\\2x_{2} x_{8} +2x_{4} x_{10} +2x_{6} x_{12} -18&=&0\\\end{array}

\begin{array}{rcl}h&=&12h_{7}+24h_{6}+22h_{5}+20h_{4}+18h_{3}+12h_{2}+6h_{1}\\e&=&x_{1} g_{37}+x_{3} g_{29}+x_{5} g_{19}+x_{6} g_{16}+x_{4} g_{15}+x_{2} g_{14}\\f&=&g_{-14}+g_{-15}+g_{-16}+g_{-19}+g_{-29}+g_{-37}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0 & 2 & 0 & 2\\ 2 & 0 & 2 & 0 & 0 & 2\\ 2 & 0 & 2 & 0 & 2 & 0\\ 1 & 0 & 1 & 0 & 1 & 0\\ 0 & 2 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 2 & 0 & 0\\ 0 & 2 & 0 & 2 & 0 & 2\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}20\\ 22\\ 24\\ 12\\ 6\\ 12\\ 18\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&12h_{7}+24h_{6}+22h_{5}+20h_{4}+18h_{3}+12h_{2}+6h_{1}\\ e&=&x_{1} g_{37}+x_{3} g_{29}+x_{5} g_{19}+x_{6} g_{16}+x_{4} g_{15}+x_{2} g_{14}\\ f&=&x_{8} g_{-14}+x_{10} g_{-15}+x_{12} g_{-16}+x_{11} g_{-19}+x_{9} g_{-29}+x_{7} g_{-37}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{7} +2x_{4} x_{10} +2x_{6} x_{12} -20&=&0\\2x_{1} x_{7} +2x_{3} x_{9} +2x_{6} x_{12} -22&=&0\\2x_{1} x_{7} +2x_{3} x_{9} +2x_{5} x_{11} -24&=&0\\x_{1} x_{7} +x_{3} x_{9} +x_{5} x_{11} -12&=&0\\2x_{2} x_{8} -6&=&0\\2x_{2} x_{8} +2x_{4} x_{10} -12&=&0\\2x_{2} x_{8} +2x_{4} x_{10} +2x_{6} x_{12} -18&=&0\\\end{array}

A^{28}_1

h-characteristic: (0, 1, 0, 1, 0, 1, 0)
Length of the weight dual to h: 56
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1:

A^{2}_2+2B^{1}_2

Containing regular semisimple subalgebra number 2:

A^{2}_3+A^{2}_2

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

3V_{6\psi}+4V_{5\psi}+4V_{4\psi}+4V_{3\psi}+4V_{2\psi}+4V_{\psi}+4V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&12h_{7}+24h_{6}+22h_{5}+20h_{4}+16h_{3}+12h_{2}+6h_{1}\\ e&=&4g_{29}+2g_{27}+3g_{21}+3g_{20}+4g_{19}+2g_{17}\\ f&=&g_{-17}+g_{-19}+g_{-20}+g_{-21}+g_{-27}+g_{-29}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&12h_{7}+24h_{6}+22h_{5}+20h_{4}+16h_{3}+12h_{2}+6h_{1}\\ [h, e]&=&8g_{29}+4g_{27}+6g_{21}+6g_{20}+8g_{19}+4g_{17}\\ [h, f]&=&-2g_{-17}-2g_{-19}-2g_{-20}-2g_{-21}-2g_{-27}-2g_{-29}\end{array}

Centralizer type:

A^{3}_1

Killing form square of Cartan element dual to ambient long root: 32
Basis of the centralizer (dimension: 4):

h_{7}-2h_{3}

g_{3}+g_{-7}

g_{5}+g_{1}-g_{-1}-g_{-5}

g_{7}+g_{-3}

Basis of centralizer intersected with cartan (dimension: 1):

1/2h_{7}-h_{3}

Cartan of centralizer (dimension: 2):

1/2h_{7}-h_{3}

g_{5}+g_{1}-g_{-1}-g_{-5}

Cartan-generating semisimple element:

1/2h_{7}-h_{3}

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0\\ 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\\ \end{pmatrix}

Characteristic polynomial ad H:

x^4-x^2

Factorization of characteristic polynomial of ad H: (x )(x )(x -1)(x +1)
Eigenvalues of ad H:

0

1

-1

4 eigenvectors of ad H:

(1,0,0,0)

(0,0,1,0)

(0,0,0,1)

(0,1,0,0)

Centralizer type: A^{3}_1
Reductive components (1 total):
Scalar product computed:

\begin{pmatrix}1/24\\ \end{pmatrix}

Simple basis of Cartan of centralizer (1 total):

h_{7}-2h_{3}

matching e:

g_{7}+g_{-3}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0\\ 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2\\ \end{pmatrix}

Linear space basis of intersection of centralizer and ambient Cartan:

h_{7}-2h_{3}

matching e:

g_{7}+g_{-3}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0\\ 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2\\ \end{pmatrix}

Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form:

\begin{pmatrix}96\\ \end{pmatrix}

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&12h_{7}+24h_{6}+22h_{5}+20h_{4}+16h_{3}+12h_{2}+6h_{1}\\ e&=&x_{3} g_{29}+x_{1} g_{27}+x_{6} g_{21}+x_{4} g_{20}+x_{5} g_{19}+x_{2} g_{17}\\ f&=&x_{8} g_{-17}+x_{11} g_{-19}+x_{10} g_{-20}+x_{12} g_{-21}+x_{7} g_{-27}+x_{9} g_{-29}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(2x_{1} x_{7} +x_{3} x_{9} +x_{5} x_{11} -12\right)h_{7}+\left(2x_{1} x_{7} +2x_{2} x_{8} +2x_{3} x_{9} +2x_{5} x_{11} -24\right)h_{6}+\left(2x_{1} x_{7} +2x_{2} x_{8} +2x_{3} x_{9} +2x_{6} x_{12} -22\right)h_{5}+\left(2x_{1} x_{7} +2x_{2} x_{8} +2x_{4} x_{10} +2x_{6} x_{12} -20\right)h_{4}+\left(2x_{1} x_{7} +2x_{4} x_{10} +2x_{6} x_{12} -16\right)h_{3}+\left(2x_{4} x_{10} +2x_{6} x_{12} -12\right)h_{2}+\left(2x_{4} x_{10} -6\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{7} +2x_{4} x_{10} +2x_{6} x_{12} -16&=&0\\2x_{1} x_{7} +2x_{2} x_{8} +2x_{4} x_{10} +2x_{6} x_{12} -20&=&0\\2x_{1} x_{7} +2x_{2} x_{8} +2x_{3} x_{9} +2x_{6} x_{12} -22&=&0\\2x_{1} x_{7} +2x_{2} x_{8} +2x_{3} x_{9} +2x_{5} x_{11} -24&=&0\\2x_{1} x_{7} +x_{3} x_{9} +x_{5} x_{11} -12&=&0\\2x_{4} x_{10} -6&=&0\\2x_{4} x_{10} +2x_{6} x_{12} -12&=&0\\\end{array}

\begin{array}{rcl}h&=&12h_{7}+24h_{6}+22h_{5}+20h_{4}+16h_{3}+12h_{2}+6h_{1}\\e&=&x_{3} g_{29}+x_{1} g_{27}+x_{6} g_{21}+x_{4} g_{20}+x_{5} g_{19}+x_{2} g_{17}\\f&=&g_{-17}+g_{-19}+g_{-20}+g_{-21}+g_{-27}+g_{-29}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0 & 2 & 0 & 2\\ 2 & 2 & 0 & 2 & 0 & 2\\ 2 & 2 & 2 & 0 & 0 & 2\\ 2 & 2 & 2 & 0 & 2 & 0\\ 2 & 0 & 1 & 0 & 1 & 0\\ 0 & 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 2\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}16\\ 20\\ 22\\ 24\\ 12\\ 6\\ 12\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&12h_{7}+24h_{6}+22h_{5}+20h_{4}+16h_{3}+12h_{2}+6h_{1}\\ e&=&x_{3} g_{29}+x_{1} g_{27}+x_{6} g_{21}+x_{4} g_{20}+x_{5} g_{19}+x_{2} g_{17}\\ f&=&x_{8} g_{-17}+x_{11} g_{-19}+x_{10} g_{-20}+x_{12} g_{-21}+x_{7} g_{-27}+x_{9} g_{-29}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{7} +2x_{4} x_{10} +2x_{6} x_{12} -16&=&0\\2x_{1} x_{7} +2x_{2} x_{8} +2x_{4} x_{10} +2x_{6} x_{12} -20&=&0\\2x_{1} x_{7} +2x_{2} x_{8} +2x_{3} x_{9} +2x_{6} x_{12} -22&=&0\\2x_{1} x_{7} +2x_{2} x_{8} +2x_{3} x_{9} +2x_{5} x_{11} -24&=&0\\2x_{1} x_{7} +x_{3} x_{9} +x_{5} x_{11} -12&=&0\\2x_{4} x_{10} -6&=&0\\2x_{4} x_{10} +2x_{6} x_{12} -12&=&0\\\end{array}

A^{23}_1

h-characteristic: (0, 2, 0, 0, 0, 0, 2)
Length of the weight dual to h: 46
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 7
Containing regular semisimple subalgebra number 1:

2B^{1}_2+3A^{1}_1

Containing regular semisimple subalgebra number 2:

A^{2}_3+3A^{1}_1

Containing regular semisimple subalgebra number 3:

2B^{1}_2+A^{2}_1+A^{1}_1

Containing regular semisimple subalgebra number 4:

A^{2}_3+A^{2}_1+A^{1}_1

Containing regular semisimple subalgebra number 5:

2C^{1}_3+A^{1}_1

Containing regular semisimple subalgebra number 6:

C^{1}_3+B^{1}_2+2A^{1}_1

Containing regular semisimple subalgebra number 7:

C^{1}_3+B^{1}_2+A^{2}_1

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

3V_{6\psi}+7V_{4\psi}+15V_{2\psi}+4V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&11h_{7}+20h_{6}+18h_{5}+16h_{4}+14h_{3}+12h_{2}+6h_{1}\\ e&=&4g_{43}+4g_{37}+g_{29}+g_{19}+3g_{9}+3g_{8}+g_{7}\\ f&=&g_{-7}+g_{-8}+g_{-9}+g_{-19}+g_{-29}+g_{-37}+g_{-43}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&11h_{7}+20h_{6}+18h_{5}+16h_{4}+14h_{3}+12h_{2}+6h_{1}\\ [h, e]&=&8g_{43}+8g_{37}+2g_{29}+2g_{19}+6g_{9}+6g_{8}+2g_{7}\\ [h, f]&=&-2g_{-7}-2g_{-8}-2g_{-9}-2g_{-19}-2g_{-29}-2g_{-37}-2g_{-43}\end{array}

Centralizer type:

A^{8}_1

Killing form square of Cartan element dual to ambient long root: 32
Basis of the centralizer (dimension: 4):

g_{3}+g_{1}-g_{-1}-g_{-3}

g_{5}-g_{-5}

g_{6}-g_{-6}

g_{12}-g_{-12}

Basis of centralizer intersected with cartan (dimension: 0):
Cartan of centralizer (dimension: 2):

g_{3}+g_{1}-g_{-1}-g_{-3}

g_{12}+g_{6}+g_{5}-g_{-5}-g_{-6}-g_{-12}

Cartan-generating semisimple element:

g_{12}+g_{6}+g_{5}+g_{3}+g_{1}-g_{-1}-g_{-3}-g_{-5}-g_{-6}-g_{-12}

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0\\ 0 & 0 & -1 & 1\\ 0 & 1 & 0 & -1\\ 0 & -1 & 1 & 0\\ \end{pmatrix}

Characteristic polynomial ad H:

x^4+3x^2

Factorization of characteristic polynomial of ad H: (x )(x )(x^2+3)
Eigenvalues of ad H:

0

\sqrt{-3}

-\sqrt{-3}

4 eigenvectors of ad H:

(1,0,0,0)

(0,1,1,1)

(0,−12−3−12,12−3−12,1)

(0,12−3−12,−12−3−12,1)

Centralizer type: A^{8}_1
Reductive components (1 total):
Scalar product computed:

\begin{pmatrix}1/64\\ \end{pmatrix}

Simple basis of Cartan of centralizer (1 total):

-2/3\sqrt{-3}g_{12}-2/3\sqrt{-3}g_{6}-2/3\sqrt{-3}g_{5}+2/3\sqrt{-3}g_{-5}+2/3\sqrt{-3}g_{-6}+2/3\sqrt{-3}g_{-12}

matching e:

g_{12}+\left(1/2\sqrt{-3}-1/2\right)g_{6}+\left(-1/2\sqrt{-3}-1/2\right)g_{5}+\left(1/2\sqrt{-3}+1/2\right)g_{-5}+\left(-1/2\sqrt{-3}+1/2\right)g_{-6}-g_{-12}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0\\ 0 & 0 & 2/3\sqrt{-3} & -2/3\sqrt{-3}\\ 0 & -2/3\sqrt{-3} & 0 & 2/3\sqrt{-3}\\ 0 & 2/3\sqrt{-3} & -2/3\sqrt{-3} & 0\\ \end{pmatrix}

Linear space basis of intersection of centralizer and ambient Cartan:

-2/3\sqrt{-3}g_{12}-2/3\sqrt{-3}g_{6}-2/3\sqrt{-3}g_{5}+2/3\sqrt{-3}g_{-5}+2/3\sqrt{-3}g_{-6}+2/3\sqrt{-3}g_{-12}

matching e:

g_{12}+\left(1/2\sqrt{-3}-1/2\right)g_{6}+\left(-1/2\sqrt{-3}-1/2\right)g_{5}+\left(1/2\sqrt{-3}+1/2\right)g_{-5}+\left(-1/2\sqrt{-3}+1/2\right)g_{-6}-g_{-12}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0\\ 0 & 0 & 2/3\sqrt{-3} & -2/3\sqrt{-3}\\ 0 & -2/3\sqrt{-3} & 0 & 2/3\sqrt{-3}\\ 0 & 2/3\sqrt{-3} & -2/3\sqrt{-3} & 0\\ \end{pmatrix}

Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form:

\begin{pmatrix}256\\ \end{pmatrix}

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&11h_{7}+20h_{6}+18h_{5}+16h_{4}+14h_{3}+12h_{2}+6h_{1}\\ e&=&x_{1} g_{43}+x_{3} g_{37}+x_{5} g_{29}+x_{6} g_{19}+x_{4} g_{9}+x_{2} g_{8}+x_{7} g_{7}\\ f&=&x_{14} g_{-7}+x_{9} g_{-8}+x_{11} g_{-9}+x_{13} g_{-19}+x_{12} g_{-29}+x_{10} g_{-37}+x_{8} g_{-43}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(x_{1} x_{8} +x_{3} x_{10} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -11\right)h_{7}+\left(2x_{1} x_{8} +2x_{3} x_{10} +2x_{5} x_{12} +2x_{6} x_{13} -20\right)h_{6}+\left(2x_{1} x_{8} +2x_{3} x_{10} +2x_{5} x_{12} -18\right)h_{5}+\left(2x_{1} x_{8} +2x_{3} x_{10} -16\right)h_{4}+\left(2x_{1} x_{8} +2x_{4} x_{11} -14\right)h_{3}+\left(2x_{2} x_{9} +2x_{4} x_{11} -12\right)h_{2}+\left(2x_{2} x_{9} -6\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{8} +2x_{4} x_{11} -14&=&0\\2x_{1} x_{8} +2x_{3} x_{10} -16&=&0\\2x_{1} x_{8} +2x_{3} x_{10} +2x_{5} x_{12} -18&=&0\\2x_{1} x_{8} +2x_{3} x_{10} +2x_{5} x_{12} +2x_{6} x_{13} -20&=&0\\x_{1} x_{8} +x_{3} x_{10} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -11&=&0\\2x_{2} x_{9} -6&=&0\\2x_{2} x_{9} +2x_{4} x_{11} -12&=&0\\\end{array}

\begin{array}{rcl}h&=&11h_{7}+20h_{6}+18h_{5}+16h_{4}+14h_{3}+12h_{2}+6h_{1}\\e&=&x_{1} g_{43}+x_{3} g_{37}+x_{5} g_{29}+x_{6} g_{19}+x_{4} g_{9}+x_{2} g_{8}+x_{7} g_{7}\\f&=&g_{-7}+g_{-8}+g_{-9}+g_{-19}+g_{-29}+g_{-37}+g_{-43}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0 & 2 & 0 & 0 & 0\\ 2 & 0 & 2 & 0 & 0 & 0 & 0\\ 2 & 0 & 2 & 0 & 2 & 0 & 0\\ 2 & 0 & 2 & 0 & 2 & 2 & 0\\ 1 & 0 & 1 & 0 & 1 & 1 & 1\\ 0 & 2 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 2 & 0 & 0 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}14\\ 16\\ 18\\ 20\\ 11\\ 6\\ 12\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&11h_{7}+20h_{6}+18h_{5}+16h_{4}+14h_{3}+12h_{2}+6h_{1}\\ e&=&x_{1} g_{43}+x_{3} g_{37}+x_{5} g_{29}+x_{6} g_{19}+x_{4} g_{9}+x_{2} g_{8}+x_{7} g_{7}\\ f&=&x_{14} g_{-7}+x_{9} g_{-8}+x_{11} g_{-9}+x_{13} g_{-19}+x_{12} g_{-29}+x_{10} g_{-37}+x_{8} g_{-43}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{8} +2x_{4} x_{11} -14&=&0\\2x_{1} x_{8} +2x_{3} x_{10} -16&=&0\\2x_{1} x_{8} +2x_{3} x_{10} +2x_{5} x_{12} -18&=&0\\2x_{1} x_{8} +2x_{3} x_{10} +2x_{5} x_{12} +2x_{6} x_{13} -20&=&0\\x_{1} x_{8} +x_{3} x_{10} +x_{5} x_{12} +x_{6} x_{13} +x_{7} x_{14} -11&=&0\\2x_{2} x_{9} -6&=&0\\2x_{2} x_{9} +2x_{4} x_{11} -12&=&0\\\end{array}

A^{22}_1

h-characteristic: (0, 2, 0, 0, 0, 1, 0)
Length of the weight dual to h: 44
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 6
Containing regular semisimple subalgebra number 1:

2B^{1}_2+2A^{1}_1

Containing regular semisimple subalgebra number 2:

A^{2}_3+2A^{1}_1

Containing regular semisimple subalgebra number 3:

2B^{1}_2+A^{2}_1

Containing regular semisimple subalgebra number 4:

A^{2}_3+A^{2}_1

Containing regular semisimple subalgebra number 5:

2C^{1}_3

Containing regular semisimple subalgebra number 6:

C^{1}_3+B^{1}_2+A^{1}_1

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

3V_{6\psi}+5V_{4\psi}+4V_{3\psi}+10V_{2\psi}+4V_{\psi}+5V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&10h_{7}+20h_{6}+18h_{5}+16h_{4}+14h_{3}+12h_{2}+6h_{1}\\ e&=&4g_{43}+4g_{37}+g_{29}+g_{19}+3g_{9}+3g_{8}\\ f&=&g_{-8}+g_{-9}+g_{-19}+g_{-29}+g_{-37}+g_{-43}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&10h_{7}+20h_{6}+18h_{5}+16h_{4}+14h_{3}+12h_{2}+6h_{1}\\ [h, e]&=&8g_{43}+8g_{37}+2g_{29}+2g_{19}+6g_{9}+6g_{8}\\ [h, f]&=&-2g_{-8}-2g_{-9}-2g_{-19}-2g_{-29}-2g_{-37}-2g_{-43}\end{array}

Centralizer type:

A_1

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&10h_{7}+20h_{6}+18h_{5}+16h_{4}+14h_{3}+12h_{2}+6h_{1}\\ e&=&x_{1} g_{43}+x_{3} g_{37}+x_{5} g_{29}+x_{6} g_{19}+x_{4} g_{9}+x_{2} g_{8}\\ f&=&x_{8} g_{-8}+x_{10} g_{-9}+x_{12} g_{-19}+x_{11} g_{-29}+x_{9} g_{-37}+x_{7} g_{-43}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(x_{1} x_{7} +x_{3} x_{9} +x_{5} x_{11} +x_{6} x_{12} -10\right)h_{7}+\left(2x_{1} x_{7} +2x_{3} x_{9} +2x_{5} x_{11} +2x_{6} x_{12} -20\right)h_{6}+\left(2x_{1} x_{7} +2x_{3} x_{9} +2x_{5} x_{11} -18\right)h_{5}+\left(2x_{1} x_{7} +2x_{3} x_{9} -16\right)h_{4}+\left(2x_{1} x_{7} +2x_{4} x_{10} -14\right)h_{3}+\left(2x_{2} x_{8} +2x_{4} x_{10} -12\right)h_{2}+\left(2x_{2} x_{8} -6\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{7} +2x_{4} x_{10} -14&=&0\\2x_{1} x_{7} +2x_{3} x_{9} -16&=&0\\2x_{1} x_{7} +2x_{3} x_{9} +2x_{5} x_{11} -18&=&0\\2x_{1} x_{7} +2x_{3} x_{9} +2x_{5} x_{11} +2x_{6} x_{12} -20&=&0\\x_{1} x_{7} +x_{3} x_{9} +x_{5} x_{11} +x_{6} x_{12} -10&=&0\\2x_{2} x_{8} -6&=&0\\2x_{2} x_{8} +2x_{4} x_{10} -12&=&0\\\end{array}

\begin{array}{rcl}h&=&10h_{7}+20h_{6}+18h_{5}+16h_{4}+14h_{3}+12h_{2}+6h_{1}\\e&=&x_{1} g_{43}+x_{3} g_{37}+x_{5} g_{29}+x_{6} g_{19}+x_{4} g_{9}+x_{2} g_{8}\\f&=&g_{-8}+g_{-9}+g_{-19}+g_{-29}+g_{-37}+g_{-43}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0 & 2 & 0 & 0\\ 2 & 0 & 2 & 0 & 0 & 0\\ 2 & 0 & 2 & 0 & 2 & 0\\ 2 & 0 & 2 & 0 & 2 & 2\\ 1 & 0 & 1 & 0 & 1 & 1\\ 0 & 2 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 2 & 0 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}14\\ 16\\ 18\\ 20\\ 10\\ 6\\ 12\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&10h_{7}+20h_{6}+18h_{5}+16h_{4}+14h_{3}+12h_{2}+6h_{1}\\ e&=&x_{1} g_{43}+x_{3} g_{37}+x_{5} g_{29}+x_{6} g_{19}+x_{4} g_{9}+x_{2} g_{8}\\ f&=&x_{8} g_{-8}+x_{10} g_{-9}+x_{12} g_{-19}+x_{11} g_{-29}+x_{9} g_{-37}+x_{7} g_{-43}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{7} +2x_{4} x_{10} -14&=&0\\2x_{1} x_{7} +2x_{3} x_{9} -16&=&0\\2x_{1} x_{7} +2x_{3} x_{9} +2x_{5} x_{11} -18&=&0\\2x_{1} x_{7} +2x_{3} x_{9} +2x_{5} x_{11} +2x_{6} x_{12} -20&=&0\\x_{1} x_{7} +x_{3} x_{9} +x_{5} x_{11} +x_{6} x_{12} -10&=&0\\2x_{2} x_{8} -6&=&0\\2x_{2} x_{8} +2x_{4} x_{10} -12&=&0\\\end{array}

A^{21}_1

h-characteristic: (0, 2, 0, 0, 1, 0, 0)
Length of the weight dual to h: 42
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1:

2B^{1}_2+A^{1}_1

Containing regular semisimple subalgebra number 2:

A^{2}_3+A^{1}_1

Containing regular semisimple subalgebra number 3:

C^{1}_3+B^{1}_2

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

3V_{6\psi}+3V_{4\psi}+8V_{3\psi}+6V_{2\psi}+4V_{\psi}+11V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&9h_{7}+18h_{6}+18h_{5}+16h_{4}+14h_{3}+12h_{2}+6h_{1}\\ e&=&4g_{43}+4g_{37}+g_{29}+3g_{9}+3g_{8}\\ f&=&g_{-8}+g_{-9}+g_{-29}+g_{-37}+g_{-43}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&9h_{7}+18h_{6}+18h_{5}+16h_{4}+14h_{3}+12h_{2}+6h_{1}\\ [h, e]&=&8g_{43}+8g_{37}+2g_{29}+6g_{9}+6g_{8}\\ [h, f]&=&-2g_{-8}-2g_{-9}-2g_{-29}-2g_{-37}-2g_{-43}\end{array}

Centralizer type:

B_2

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&9h_{7}+18h_{6}+18h_{5}+16h_{4}+14h_{3}+12h_{2}+6h_{1}\\ e&=&x_{1} g_{43}+x_{3} g_{37}+x_{5} g_{29}+x_{4} g_{9}+x_{2} g_{8}\\ f&=&x_{7} g_{-8}+x_{9} g_{-9}+x_{10} g_{-29}+x_{8} g_{-37}+x_{6} g_{-43}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(x_{1} x_{6} +x_{3} x_{8} +x_{5} x_{10} -9\right)h_{7}+\left(2x_{1} x_{6} +2x_{3} x_{8} +2x_{5} x_{10} -18\right)h_{6}+\left(2x_{1} x_{6} +2x_{3} x_{8} +2x_{5} x_{10} -18\right)h_{5}+\left(2x_{1} x_{6} +2x_{3} x_{8} -16\right)h_{4}+\left(2x_{1} x_{6} +2x_{4} x_{9} -14\right)h_{3}+\left(2x_{2} x_{7} +2x_{4} x_{9} -12\right)h_{2}+\left(2x_{2} x_{7} -6\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{6} +2x_{4} x_{9} -14&=&0\\2x_{1} x_{6} +2x_{3} x_{8} -16&=&0\\2x_{1} x_{6} +2x_{3} x_{8} +2x_{5} x_{10} -18&=&0\\2x_{1} x_{6} +2x_{3} x_{8} +2x_{5} x_{10} -18&=&0\\x_{1} x_{6} +x_{3} x_{8} +x_{5} x_{10} -9&=&0\\2x_{2} x_{7} -6&=&0\\2x_{2} x_{7} +2x_{4} x_{9} -12&=&0\\\end{array}

\begin{array}{rcl}h&=&9h_{7}+18h_{6}+18h_{5}+16h_{4}+14h_{3}+12h_{2}+6h_{1}\\e&=&x_{1} g_{43}+x_{3} g_{37}+x_{5} g_{29}+x_{4} g_{9}+x_{2} g_{8}\\f&=&g_{-8}+g_{-9}+g_{-29}+g_{-37}+g_{-43}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0 & 2 & 0\\ 2 & 0 & 2 & 0 & 0\\ 2 & 0 & 2 & 0 & 2\\ 2 & 0 & 2 & 0 & 2\\ 1 & 0 & 1 & 0 & 1\\ 0 & 2 & 0 & 0 & 0\\ 0 & 2 & 0 & 2 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}14\\ 16\\ 18\\ 18\\ 9\\ 6\\ 12\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&9h_{7}+18h_{6}+18h_{5}+16h_{4}+14h_{3}+12h_{2}+6h_{1}\\ e&=&x_{1} g_{43}+x_{3} g_{37}+x_{5} g_{29}+x_{4} g_{9}+x_{2} g_{8}\\ f&=&x_{7} g_{-8}+x_{9} g_{-9}+x_{10} g_{-29}+x_{8} g_{-37}+x_{6} g_{-43}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{6} +2x_{4} x_{9} -14&=&0\\2x_{1} x_{6} +2x_{3} x_{8} -16&=&0\\2x_{1} x_{6} +2x_{3} x_{8} +2x_{5} x_{10} -18&=&0\\2x_{1} x_{6} +2x_{3} x_{8} +2x_{5} x_{10} -18&=&0\\x_{1} x_{6} +x_{3} x_{8} +x_{5} x_{10} -9&=&0\\2x_{2} x_{7} -6&=&0\\2x_{2} x_{7} +2x_{4} x_{9} -12&=&0\\\end{array}

A^{20}_1

h-characteristic: (1, 0, 1, 0, 0, 1, 0)
Length of the weight dual to h: 40
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1:

A^{2}_2+B^{1}_2+2A^{1}_1

Containing regular semisimple subalgebra number 2:

A^{2}_2+B^{1}_2+A^{2}_1

Containing regular semisimple subalgebra number 3:

C^{1}_3+A^{2}_2+A^{1}_1

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

V_{6\psi}+2V_{5\psi}+5V_{4\psi}+6V_{3\psi}+7V_{2\psi}+6V_{\psi}+4V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&10h_{7}+20h_{6}+18h_{5}+16h_{4}+14h_{3}+10h_{2}+6h_{1}\\ e&=&4g_{37}+2g_{31}+g_{29}+2g_{22}+g_{19}+3g_{14}\\ f&=&g_{-14}+g_{-19}+g_{-22}+g_{-29}+g_{-31}+g_{-37}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&10h_{7}+20h_{6}+18h_{5}+16h_{4}+14h_{3}+10h_{2}+6h_{1}\\ [h, e]&=&8g_{37}+4g_{31}+2g_{29}+4g_{22}+2g_{19}+6g_{14}\\ [h, f]&=&-2g_{-14}-2g_{-19}-2g_{-22}-2g_{-29}-2g_{-31}-2g_{-37}\end{array}

Centralizer type:

A^{3}_1

Killing form square of Cartan element dual to ambient long root: 32
Basis of the centralizer (dimension: 4):

h_{7}-2h_{2}

g_{2}+g_{-7}

g_{5}-g_{-5}

g_{7}+g_{-2}

Basis of centralizer intersected with cartan (dimension: 1):

1/2h_{7}-h_{2}

Cartan of centralizer (dimension: 2):

1/2h_{7}-h_{2}

g_{5}-g_{-5}

Cartan-generating semisimple element:

1/2h_{7}-h_{2}

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0\\ 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1\\ \end{pmatrix}

Characteristic polynomial ad H:

x^4-x^2

Factorization of characteristic polynomial of ad H: (x )(x )(x -1)(x +1)
Eigenvalues of ad H:

0

1

-1

4 eigenvectors of ad H:

(1,0,0,0)

(0,0,1,0)

(0,0,0,1)

(0,1,0,0)

Centralizer type: A^{3}_1
Reductive components (1 total):
Scalar product computed:

\begin{pmatrix}1/24\\ \end{pmatrix}

Simple basis of Cartan of centralizer (1 total):

h_{7}-2h_{2}

matching e:

g_{7}+g_{-2}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0\\ 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2\\ \end{pmatrix}

Linear space basis of intersection of centralizer and ambient Cartan:

h_{7}-2h_{2}

matching e:

g_{7}+g_{-2}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0\\ 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2\\ \end{pmatrix}

Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form:

\begin{pmatrix}96\\ \end{pmatrix}

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&10h_{7}+20h_{6}+18h_{5}+16h_{4}+14h_{3}+10h_{2}+6h_{1}\\ e&=&x_{3} g_{37}+x_{1} g_{31}+x_{5} g_{29}+x_{2} g_{22}+x_{6} g_{19}+x_{4} g_{14}\\ f&=&x_{10} g_{-14}+x_{12} g_{-19}+x_{8} g_{-22}+x_{11} g_{-29}+x_{7} g_{-31}+x_{9} g_{-37}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(2x_{1} x_{7} +x_{3} x_{9} +x_{5} x_{11} +x_{6} x_{12} -10\right)h_{7}+\left(2x_{1} x_{7} +2x_{2} x_{8} +2x_{3} x_{9} +2x_{5} x_{11} +2x_{6} x_{12} -20\right)h_{6}+\left(2x_{1} x_{7} +2x_{2} x_{8} +2x_{3} x_{9} +2x_{5} x_{11} -18\right)h_{5}+\left(2x_{1} x_{7} +2x_{2} x_{8} +2x_{3} x_{9} -16\right)h_{4}+\left(2x_{1} x_{7} +2x_{2} x_{8} +2x_{4} x_{10} -14\right)h_{3}+\left(2x_{1} x_{7} +2x_{4} x_{10} -10\right)h_{2}+\left(2x_{4} x_{10} -6\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{7} +2x_{4} x_{10} -10&=&0\\2x_{1} x_{7} +2x_{2} x_{8} +2x_{4} x_{10} -14&=&0\\2x_{1} x_{7} +2x_{2} x_{8} +2x_{3} x_{9} -16&=&0\\2x_{1} x_{7} +2x_{2} x_{8} +2x_{3} x_{9} +2x_{5} x_{11} -18&=&0\\2x_{1} x_{7} +2x_{2} x_{8} +2x_{3} x_{9} +2x_{5} x_{11} +2x_{6} x_{12} -20&=&0\\2x_{1} x_{7} +x_{3} x_{9} +x_{5} x_{11} +x_{6} x_{12} -10&=&0\\2x_{4} x_{10} -6&=&0\\\end{array}

\begin{array}{rcl}h&=&10h_{7}+20h_{6}+18h_{5}+16h_{4}+14h_{3}+10h_{2}+6h_{1}\\e&=&x_{3} g_{37}+x_{1} g_{31}+x_{5} g_{29}+x_{2} g_{22}+x_{6} g_{19}+x_{4} g_{14}\\f&=&g_{-14}+g_{-19}+g_{-22}+g_{-29}+g_{-31}+g_{-37}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0 & 2 & 0 & 0\\ 2 & 2 & 0 & 2 & 0 & 0\\ 2 & 2 & 2 & 0 & 0 & 0\\ 2 & 2 & 2 & 0 & 2 & 0\\ 2 & 2 & 2 & 0 & 2 & 2\\ 2 & 0 & 1 & 0 & 1 & 1\\ 0 & 0 & 0 & 2 & 0 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}10\\ 14\\ 16\\ 18\\ 20\\ 10\\ 6\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&10h_{7}+20h_{6}+18h_{5}+16h_{4}+14h_{3}+10h_{2}+6h_{1}\\ e&=&x_{3} g_{37}+x_{1} g_{31}+x_{5} g_{29}+x_{2} g_{22}+x_{6} g_{19}+x_{4} g_{14}\\ f&=&x_{10} g_{-14}+x_{12} g_{-19}+x_{8} g_{-22}+x_{11} g_{-29}+x_{7} g_{-31}+x_{9} g_{-37}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{7} +2x_{4} x_{10} -10&=&0\\2x_{1} x_{7} +2x_{2} x_{8} +2x_{4} x_{10} -14&=&0\\2x_{1} x_{7} +2x_{2} x_{8} +2x_{3} x_{9} -16&=&0\\2x_{1} x_{7} +2x_{2} x_{8} +2x_{3} x_{9} +2x_{5} x_{11} -18&=&0\\2x_{1} x_{7} +2x_{2} x_{8} +2x_{3} x_{9} +2x_{5} x_{11} +2x_{6} x_{12} -20&=&0\\2x_{1} x_{7} +x_{3} x_{9} +x_{5} x_{11} +x_{6} x_{12} -10&=&0\\2x_{4} x_{10} -6&=&0\\\end{array}

A^{20}_1

h-characteristic: (0, 2, 0, 1, 0, 0, 0)
Length of the weight dual to h: 40
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1:

2B^{1}_2

Containing regular semisimple subalgebra number 2:

A^{2}_3

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

3V_{6\psi}+V_{4\psi}+12V_{3\psi}+3V_{2\psi}+22V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&8h_{7}+16h_{6}+16h_{5}+16h_{4}+14h_{3}+12h_{2}+6h_{1}\\ e&=&4g_{43}+4g_{37}+3g_{9}+3g_{8}\\ f&=&g_{-8}+g_{-9}+g_{-37}+g_{-43}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&8h_{7}+16h_{6}+16h_{5}+16h_{4}+14h_{3}+12h_{2}+6h_{1}\\ [h, e]&=&8g_{43}+8g_{37}+6g_{9}+6g_{8}\\ [h, f]&=&-2g_{-8}-2g_{-9}-2g_{-37}-2g_{-43}\end{array}

Centralizer type:

C_3

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&8h_{7}+16h_{6}+16h_{5}+16h_{4}+14h_{3}+12h_{2}+6h_{1}\\ e&=&x_{1} g_{43}+x_{3} g_{37}+x_{4} g_{9}+x_{2} g_{8}\\ f&=&x_{6} g_{-8}+x_{8} g_{-9}+x_{7} g_{-37}+x_{5} g_{-43}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(x_{1} x_{5} +x_{3} x_{7} -8\right)h_{7}+\left(2x_{1} x_{5} +2x_{3} x_{7} -16\right)h_{6}+\left(2x_{1} x_{5} +2x_{3} x_{7} -16\right)h_{5}+\left(2x_{1} x_{5} +2x_{3} x_{7} -16\right)h_{4}+\left(2x_{1} x_{5} +2x_{4} x_{8} -14\right)h_{3}+\left(2x_{2} x_{6} +2x_{4} x_{8} -12\right)h_{2}+\left(2x_{2} x_{6} -6\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{5} +2x_{4} x_{8} -14&=&0\\2x_{1} x_{5} +2x_{3} x_{7} -16&=&0\\2x_{1} x_{5} +2x_{3} x_{7} -16&=&0\\2x_{1} x_{5} +2x_{3} x_{7} -16&=&0\\x_{1} x_{5} +x_{3} x_{7} -8&=&0\\2x_{2} x_{6} -6&=&0\\2x_{2} x_{6} +2x_{4} x_{8} -12&=&0\\\end{array}

\begin{array}{rcl}h&=&8h_{7}+16h_{6}+16h_{5}+16h_{4}+14h_{3}+12h_{2}+6h_{1}\\e&=&x_{1} g_{43}+x_{3} g_{37}+x_{4} g_{9}+x_{2} g_{8}\\f&=&g_{-8}+g_{-9}+g_{-37}+g_{-43}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0 & 2\\ 2 & 0 & 2 & 0\\ 2 & 0 & 2 & 0\\ 2 & 0 & 2 & 0\\ 1 & 0 & 1 & 0\\ 0 & 2 & 0 & 0\\ 0 & 2 & 0 & 2\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}14\\ 16\\ 16\\ 16\\ 8\\ 6\\ 12\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&8h_{7}+16h_{6}+16h_{5}+16h_{4}+14h_{3}+12h_{2}+6h_{1}\\ e&=&x_{1} g_{43}+x_{3} g_{37}+x_{4} g_{9}+x_{2} g_{8}\\ f&=&x_{6} g_{-8}+x_{8} g_{-9}+x_{7} g_{-37}+x_{5} g_{-43}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{5} +2x_{4} x_{8} -14&=&0\\2x_{1} x_{5} +2x_{3} x_{7} -16&=&0\\2x_{1} x_{5} +2x_{3} x_{7} -16&=&0\\2x_{1} x_{5} +2x_{3} x_{7} -16&=&0\\x_{1} x_{5} +x_{3} x_{7} -8&=&0\\2x_{2} x_{6} -6&=&0\\2x_{2} x_{6} +2x_{4} x_{8} -12&=&0\\\end{array}

A^{19}_1

h-characteristic: (1, 0, 1, 0, 1, 0, 0)
Length of the weight dual to h: 38
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1:

A^{2}_2+B^{1}_2+A^{1}_1

Containing regular semisimple subalgebra number 2:

C^{1}_3+A^{2}_2

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

V_{6\psi}+2V_{5\psi}+4V_{4\psi}+6V_{3\psi}+8V_{2\psi}+6V_{\psi}+6V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&9h_{7}+18h_{6}+18h_{5}+16h_{4}+14h_{3}+10h_{2}+6h_{1}\\ e&=&4g_{37}+2g_{35}+g_{29}+2g_{16}+3g_{14}\\ f&=&g_{-14}+g_{-16}+g_{-29}+g_{-35}+g_{-37}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&9h_{7}+18h_{6}+18h_{5}+16h_{4}+14h_{3}+10h_{2}+6h_{1}\\ [h, e]&=&8g_{37}+4g_{35}+2g_{29}+4g_{16}+6g_{14}\\ [h, f]&=&-2g_{-14}-2g_{-16}-2g_{-29}-2g_{-35}-2g_{-37}\end{array}

Centralizer type:

A^{3}_1+A_1

Killing form square of Cartan element dual to ambient long root: 32
Basis of the centralizer (dimension: 6):

g_{-7}

h_{6}-h_{2}

h_{7}

g_{2}+g_{-19}

g_{7}

g_{19}+g_{-2}

Basis of centralizer intersected with cartan (dimension: 2):

h_{6}-h_{2}

-h_{7}

Cartan of centralizer (dimension: 2):

h_{6}-h_{2}

-h_{7}

Cartan-generating semisimple element:

-h_{7}+h_{6}-h_{2}

adjoint action:

\begin{pmatrix}3 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & -3 & 0\\ 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}

Characteristic polynomial ad H:

x^6-10x^4+9x^2

Factorization of characteristic polynomial of ad H: (x )(x )(x -3)(x -1)(x +1)(x +3)
Eigenvalues of ad H:

0

3

1

-1

-3

6 eigenvectors of ad H:

(0,1,0,0,0,0)

(0,0,1,0,0,0)

(1,0,0,0,0,0)

(0,0,0,0,0,1)

(0,0,0,1,0,0)

(0,0,0,0,1,0)

Centralizer type: A^{3}_1+A^{1}_1
Reductive components (2 total):
Scalar product computed:

\begin{pmatrix}1/8\\ \end{pmatrix}

Simple basis of Cartan of centralizer (1 total):

-h_{7}

matching e:

g_{-7}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

Linear space basis of intersection of centralizer and ambient Cartan:

-h_{7}

matching e:

g_{-7}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form:

\begin{pmatrix}32\\ \end{pmatrix}

Scalar product computed:

\begin{pmatrix}1/24\\ \end{pmatrix}

Simple basis of Cartan of centralizer (1 total):

h_{7}+2h_{6}-2h_{2}

matching e:

g_{19}+g_{-2}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}

Linear space basis of intersection of centralizer and ambient Cartan:

h_{7}+2h_{6}-2h_{2}

matching e:

g_{19}+g_{-2}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}

Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form:

\begin{pmatrix}96\\ \end{pmatrix}

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&9h_{7}+18h_{6}+18h_{5}+16h_{4}+14h_{3}+10h_{2}+6h_{1}\\ e&=&x_{3} g_{37}+x_{1} g_{35}+x_{5} g_{29}+x_{2} g_{16}+x_{4} g_{14}\\ f&=&x_{9} g_{-14}+x_{7} g_{-16}+x_{10} g_{-29}+x_{6} g_{-35}+x_{8} g_{-37}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(2x_{1} x_{6} +x_{3} x_{8} +x_{5} x_{10} -9\right)h_{7}+\left(4x_{1} x_{6} +2x_{3} x_{8} +2x_{5} x_{10} -18\right)h_{6}+\left(2x_{1} x_{6} +2x_{2} x_{7} +2x_{3} x_{8} +2x_{5} x_{10} -18\right)h_{5}+\left(2x_{1} x_{6} +2x_{2} x_{7} +2x_{3} x_{8} -16\right)h_{4}+\left(2x_{1} x_{6} +2x_{2} x_{7} +2x_{4} x_{9} -14\right)h_{3}+\left(2x_{1} x_{6} +2x_{4} x_{9} -10\right)h_{2}+\left(2x_{4} x_{9} -6\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{6} +2x_{4} x_{9} -10&=&0\\2x_{1} x_{6} +2x_{2} x_{7} +2x_{4} x_{9} -14&=&0\\2x_{1} x_{6} +2x_{2} x_{7} +2x_{3} x_{8} -16&=&0\\2x_{1} x_{6} +2x_{2} x_{7} +2x_{3} x_{8} +2x_{5} x_{10} -18&=&0\\4x_{1} x_{6} +2x_{3} x_{8} +2x_{5} x_{10} -18&=&0\\2x_{1} x_{6} +x_{3} x_{8} +x_{5} x_{10} -9&=&0\\2x_{4} x_{9} -6&=&0\\\end{array}

\begin{array}{rcl}h&=&9h_{7}+18h_{6}+18h_{5}+16h_{4}+14h_{3}+10h_{2}+6h_{1}\\e&=&x_{3} g_{37}+x_{1} g_{35}+x_{5} g_{29}+x_{2} g_{16}+x_{4} g_{14}\\f&=&g_{-14}+g_{-16}+g_{-29}+g_{-35}+g_{-37}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0 & 2 & 0\\ 2 & 2 & 0 & 2 & 0\\ 2 & 2 & 2 & 0 & 0\\ 2 & 2 & 2 & 0 & 2\\ 4 & 0 & 2 & 0 & 2\\ 2 & 0 & 1 & 0 & 1\\ 0 & 0 & 0 & 2 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}10\\ 14\\ 16\\ 18\\ 18\\ 9\\ 6\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&9h_{7}+18h_{6}+18h_{5}+16h_{4}+14h_{3}+10h_{2}+6h_{1}\\ e&=&x_{3} g_{37}+x_{1} g_{35}+x_{5} g_{29}+x_{2} g_{16}+x_{4} g_{14}\\ f&=&x_{9} g_{-14}+x_{7} g_{-16}+x_{10} g_{-29}+x_{6} g_{-35}+x_{8} g_{-37}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{6} +2x_{4} x_{9} -10&=&0\\2x_{1} x_{6} +2x_{2} x_{7} +2x_{4} x_{9} -14&=&0\\2x_{1} x_{6} +2x_{2} x_{7} +2x_{3} x_{8} -16&=&0\\2x_{1} x_{6} +2x_{2} x_{7} +2x_{3} x_{8} +2x_{5} x_{10} -18&=&0\\4x_{1} x_{6} +2x_{3} x_{8} +2x_{5} x_{10} -18&=&0\\2x_{1} x_{6} +x_{3} x_{8} +x_{5} x_{10} -9&=&0\\2x_{4} x_{9} -6&=&0\\\end{array}

A^{18}_1

h-characteristic: (1, 0, 1, 1, 0, 0, 0)
Length of the weight dual to h: 36
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1:

A^{2}_2+B^{1}_2

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

V_{6\psi}+2V_{5\psi}+3V_{4\psi}+6V_{3\psi}+10V_{2\psi}+2V_{\psi}+13V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&8h_{7}+16h_{6}+16h_{5}+16h_{4}+14h_{3}+10h_{2}+6h_{1}\\ e&=&2g_{39}+4g_{37}+3g_{14}+2g_{10}\\ f&=&g_{-10}+g_{-14}+g_{-37}+g_{-39}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&8h_{7}+16h_{6}+16h_{5}+16h_{4}+14h_{3}+10h_{2}+6h_{1}\\ [h, e]&=&4g_{39}+8g_{37}+6g_{14}+4g_{10}\\ [h, f]&=&-2g_{-10}-2g_{-14}-2g_{-37}-2g_{-39}\end{array}

Centralizer type:

B_2+A^{3}_1

Killing form square of Cartan element dual to ambient long root: 32
Basis of the centralizer (dimension: 13):

g_{-19}

g_{-13}

g_{-7}

g_{-6}

h_{5}-h_{2}

h_{6}

h_{7}

g_{2}+g_{-29}

g_{6}

g_{7}

g_{13}

g_{19}

g_{29}+g_{-2}

Basis of centralizer intersected with cartan (dimension: 3):

h_{5}-h_{2}

-h_{7}

-h_{6}

Cartan of centralizer (dimension: 3):

-h_{7}

h_{5}-h_{2}

-h_{6}

Cartan-generating semisimple element:

-h_{7}-h_{6}+h_{5}-h_{2}

adjoint action:

\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 3/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1/2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -3/2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}

Characteristic polynomial ad H:

x^{13}-17/2x^{11}+393/16x^9-239/8x^7+241/16x^5-9/4x^3

Factorization of characteristic polynomial of ad H: 1/16(x )(x )(x )(x -2)(x -1)(x -1)(x +1)(x +1)(x +2)(2x -3)(2x -1)(2x +1)(2x +3)
Eigenvalues of ad H:

0

2

1

-1

-2

3/2

1/2

-1/2

-3/2

13 eigenvectors of ad H:

(0,0,0,0,1,0,0,0,0,0,0,0,0)

(0,0,0,0,0,1,0,0,0,0,0,0,0)

(0,0,0,0,0,0,1,0,0,0,0,0,0)

(1,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,1,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,1)

(0,0,0,0,0,0,0,1,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,1,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,1,0)

(0,1,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,1,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,1,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,1,0,0)

Centralizer type: B^{1}_2+A^{3}_1
Reductive components (2 total):
Scalar product computed:

\begin{pmatrix}1/8 & -1/16\\ -1/16 & 1/16\\ \end{pmatrix}

Simple basis of Cartan of centralizer (2 total):

-h_{7}

matching e:

g_{-7}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

-2h_{6}

matching e:

g_{-6}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

Linear space basis of intersection of centralizer and ambient Cartan:

-h_{7}

matching e:

g_{-7}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

-2h_{6}

matching e:

g_{-6}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

Elements in Cartan dual to root system: (1, 1), (-1, -1), (1, 0), (-1, 0), (2, 1), (-2, -1), (0, 1), (0, -1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form:

\begin{pmatrix}32 & -32\\ -32 & 64\\ \end{pmatrix}

Scalar product computed:

\begin{pmatrix}1/24\\ \end{pmatrix}

Simple basis of Cartan of centralizer (1 total):

h_{7}+2h_{6}+2h_{5}-2h_{2}

matching e:

g_{29}+g_{-2}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}

Linear space basis of intersection of centralizer and ambient Cartan:

h_{7}+2h_{6}+2h_{5}-2h_{2}

matching e:

g_{29}+g_{-2}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}

Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form:

\begin{pmatrix}96\\ \end{pmatrix}

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&8h_{7}+16h_{6}+16h_{5}+16h_{4}+14h_{3}+10h_{2}+6h_{1}\\ e&=&x_{1} g_{39}+x_{3} g_{37}+x_{4} g_{14}+x_{2} g_{10}\\ f&=&x_{6} g_{-10}+x_{8} g_{-14}+x_{7} g_{-37}+x_{5} g_{-39}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(2x_{1} x_{5} +x_{3} x_{7} -8\right)h_{7}+\left(4x_{1} x_{5} +2x_{3} x_{7} -16\right)h_{6}+\left(4x_{1} x_{5} +2x_{3} x_{7} -16\right)h_{5}+\left(2x_{1} x_{5} +2x_{2} x_{6} +2x_{3} x_{7} -16\right)h_{4}+\left(2x_{1} x_{5} +2x_{2} x_{6} +2x_{4} x_{8} -14\right)h_{3}+\left(2x_{1} x_{5} +2x_{4} x_{8} -10\right)h_{2}+\left(2x_{4} x_{8} -6\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{5} +2x_{4} x_{8} -10&=&0\\2x_{1} x_{5} +2x_{2} x_{6} +2x_{4} x_{8} -14&=&0\\2x_{1} x_{5} +2x_{2} x_{6} +2x_{3} x_{7} -16&=&0\\4x_{1} x_{5} +2x_{3} x_{7} -16&=&0\\4x_{1} x_{5} +2x_{3} x_{7} -16&=&0\\2x_{1} x_{5} +x_{3} x_{7} -8&=&0\\2x_{4} x_{8} -6&=&0\\\end{array}

\begin{array}{rcl}h&=&8h_{7}+16h_{6}+16h_{5}+16h_{4}+14h_{3}+10h_{2}+6h_{1}\\e&=&x_{1} g_{39}+x_{3} g_{37}+x_{4} g_{14}+x_{2} g_{10}\\f&=&g_{-10}+g_{-14}+g_{-37}+g_{-39}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0 & 2\\ 2 & 2 & 0 & 2\\ 2 & 2 & 2 & 0\\ 4 & 0 & 2 & 0\\ 4 & 0 & 2 & 0\\ 2 & 0 & 1 & 0\\ 0 & 0 & 0 & 2\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}10\\ 14\\ 16\\ 16\\ 16\\ 8\\ 6\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&8h_{7}+16h_{6}+16h_{5}+16h_{4}+14h_{3}+10h_{2}+6h_{1}\\ e&=&x_{1} g_{39}+x_{3} g_{37}+x_{4} g_{14}+x_{2} g_{10}\\ f&=&x_{6} g_{-10}+x_{8} g_{-14}+x_{7} g_{-37}+x_{5} g_{-39}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{5} +2x_{4} x_{8} -10&=&0\\2x_{1} x_{5} +2x_{2} x_{6} +2x_{4} x_{8} -14&=&0\\2x_{1} x_{5} +2x_{2} x_{6} +2x_{3} x_{7} -16&=&0\\4x_{1} x_{5} +2x_{3} x_{7} -16&=&0\\4x_{1} x_{5} +2x_{3} x_{7} -16&=&0\\2x_{1} x_{5} +x_{3} x_{7} -8&=&0\\2x_{4} x_{8} -6&=&0\\\end{array}

A^{17}_1

h-characteristic: (0, 0, 0, 1, 1, 0, 0)
Length of the weight dual to h: 34
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1:

2A^{2}_2+A^{1}_1

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

10V_{4\psi}+4V_{3\psi}+7V_{2\psi}+4V_{\psi}+10V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&9h_{7}+18h_{6}+18h_{5}+16h_{4}+12h_{3}+8h_{2}+4h_{1}\\ e&=&2g_{38}+g_{29}+2g_{27}+2g_{21}+2g_{17}\\ f&=&g_{-17}+g_{-21}+g_{-27}+g_{-29}+g_{-38}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&9h_{7}+18h_{6}+18h_{5}+16h_{4}+12h_{3}+8h_{2}+4h_{1}\\ [h, e]&=&4g_{38}+2g_{29}+4g_{27}+4g_{21}+4g_{17}\\ [h, f]&=&-2g_{-17}-2g_{-21}-2g_{-27}-2g_{-29}-2g_{-38}\end{array}

Centralizer type:

B^{3}_2

Killing form square of Cartan element dual to ambient long root: 32
Basis of the centralizer (dimension: 10):

h_{6}+h_{3}-h_{1}

h_{7}-2h_{3}

g_{1}+g_{-19}

g_{3}+g_{-7}

g_{7}+g_{-3}

g_{8}-g_{-6}-g_{-9}

g_{9}+g_{6}-g_{-8}

g_{13}+g_{2}+g_{-14}

g_{14}+g_{-2}+g_{-13}

g_{19}+g_{-1}

Basis of centralizer intersected with cartan (dimension: 2):

1/2h_{7}-h_{3}

1/2h_{7}+h_{6}-h_{1}

Cartan of centralizer (dimension: 2):

1/2h_{7}-h_{3}

1/2h_{7}+h_{6}-h_{1}

Cartan-generating semisimple element:

4h_{7}+7h_{6}-h_{3}-7h_{1}

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -7 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -3 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 3 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -4 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 7\\ \end{pmatrix}

Characteristic polynomial ad H:

x^{10}-75x^8+1443x^6-8425x^4+7056x^2

Factorization of characteristic polynomial of ad H: (x )(x )(x -7)(x -4)(x -3)(x -1)(x +1)(x +3)(x +4)(x +7)
Eigenvalues of ad H:

0

7

4

3

1

-1

-3

-4

-7

10 eigenvectors of ad H:

(1,0,0,0,0,0,0,0,0,0)

(0,1,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,1)

(0,0,0,0,0,0,0,1,0,0)

(0,0,0,0,0,0,1,0,0,0)

(0,0,0,0,1,0,0,0,0,0)

(0,0,0,1,0,0,0,0,0,0)

(0,0,0,0,0,1,0,0,0,0)

(0,0,0,0,0,0,0,0,1,0)

(0,0,1,0,0,0,0,0,0,0)

Centralizer type: B^{3}_2
Reductive components (1 total):
Scalar product computed:

\begin{pmatrix}1/48 & -1/48\\ -1/48 & 1/24\\ \end{pmatrix}

Simple basis of Cartan of centralizer (2 total):

2h_{6}+2h_{3}-2h_{1}

matching e:

g_{9}+g_{6}-g_{-8}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}

h_{7}-2h_{3}

matching e:

g_{7}+g_{-3}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

Linear space basis of intersection of centralizer and ambient Cartan:

2h_{6}+2h_{3}-2h_{1}

matching e:

g_{9}+g_{6}-g_{-8}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}

h_{7}-2h_{3}

matching e:

g_{7}+g_{-3}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

Elements in Cartan dual to root system: (1, 1), (-1, -1), (1, 2), (-1, -2), (1, 0), (-1, 0), (0, 1), (0, -1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form:

\begin{pmatrix}192 & -96\\ -96 & 96\\ \end{pmatrix}

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&9h_{7}+18h_{6}+18h_{5}+16h_{4}+12h_{3}+8h_{2}+4h_{1}\\ e&=&x_{1} g_{38}+x_{5} g_{29}+x_{3} g_{27}+x_{2} g_{21}+x_{4} g_{17}\\ f&=&x_{9} g_{-17}+x_{7} g_{-21}+x_{8} g_{-27}+x_{10} g_{-29}+x_{6} g_{-38}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(2x_{1} x_{6} +2x_{3} x_{8} +x_{5} x_{10} -9\right)h_{7}+\left(4x_{1} x_{6} +2x_{3} x_{8} +2x_{4} x_{9} +2x_{5} x_{10} -18\right)h_{6}+\left(2x_{1} x_{6} +2x_{2} x_{7} +2x_{3} x_{8} +2x_{4} x_{9} +2x_{5} x_{10} -18\right)h_{5}+\left(2x_{1} x_{6} +2x_{2} x_{7} +2x_{3} x_{8} +2x_{4} x_{9} -16\right)h_{4}+\left(2x_{1} x_{6} +2x_{2} x_{7} +2x_{3} x_{8} -12\right)h_{3}+\left(2x_{1} x_{6} +2x_{2} x_{7} -8\right)h_{2}+\left(2x_{1} x_{6} -4\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{6} -4&=&0\\2x_{1} x_{6} +2x_{2} x_{7} -8&=&0\\2x_{1} x_{6} +2x_{2} x_{7} +2x_{3} x_{8} -12&=&0\\2x_{1} x_{6} +2x_{2} x_{7} +2x_{3} x_{8} +2x_{4} x_{9} -16&=&0\\2x_{1} x_{6} +2x_{2} x_{7} +2x_{3} x_{8} +2x_{4} x_{9} +2x_{5} x_{10} -18&=&0\\4x_{1} x_{6} +2x_{3} x_{8} +2x_{4} x_{9} +2x_{5} x_{10} -18&=&0\\2x_{1} x_{6} +2x_{3} x_{8} +x_{5} x_{10} -9&=&0\\\end{array}

\begin{array}{rcl}h&=&9h_{7}+18h_{6}+18h_{5}+16h_{4}+12h_{3}+8h_{2}+4h_{1}\\e&=&x_{1} g_{38}+x_{5} g_{29}+x_{3} g_{27}+x_{2} g_{21}+x_{4} g_{17}\\f&=&g_{-17}+g_{-21}+g_{-27}+g_{-29}+g_{-38}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0 & 0 & 0\\ 2 & 2 & 0 & 0 & 0\\ 2 & 2 & 2 & 0 & 0\\ 2 & 2 & 2 & 2 & 0\\ 2 & 2 & 2 & 2 & 2\\ 4 & 0 & 2 & 2 & 2\\ 2 & 0 & 2 & 0 & 1\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}4\\ 8\\ 12\\ 16\\ 18\\ 18\\ 9\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&9h_{7}+18h_{6}+18h_{5}+16h_{4}+12h_{3}+8h_{2}+4h_{1}\\ e&=&x_{1} g_{38}+x_{5} g_{29}+x_{3} g_{27}+x_{2} g_{21}+x_{4} g_{17}\\ f&=&x_{9} g_{-17}+x_{7} g_{-21}+x_{8} g_{-27}+x_{10} g_{-29}+x_{6} g_{-38}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{6} -4&=&0\\2x_{1} x_{6} +2x_{2} x_{7} -8&=&0\\2x_{1} x_{6} +2x_{2} x_{7} +2x_{3} x_{8} -12&=&0\\2x_{1} x_{6} +2x_{2} x_{7} +2x_{3} x_{8} +2x_{4} x_{9} -16&=&0\\2x_{1} x_{6} +2x_{2} x_{7} +2x_{3} x_{8} +2x_{4} x_{9} +2x_{5} x_{10} -18&=&0\\4x_{1} x_{6} +2x_{3} x_{8} +2x_{4} x_{9} +2x_{5} x_{10} -18&=&0\\2x_{1} x_{6} +2x_{3} x_{8} +x_{5} x_{10} -9&=&0\\\end{array}

A^{16}_1

h-characteristic: (0, 0, 0, 2, 0, 0, 0)
Length of the weight dual to h: 32
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1:

2A^{2}_2

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

10V_{4\psi}+14V_{2\psi}+13V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&8h_{7}+16h_{6}+16h_{5}+16h_{4}+12h_{3}+8h_{2}+4h_{1}\\ e&=&2g_{41}+2g_{32}+2g_{15}+2g_{11}\\ f&=&g_{-11}+g_{-15}+g_{-32}+g_{-41}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&8h_{7}+16h_{6}+16h_{5}+16h_{4}+12h_{3}+8h_{2}+4h_{1}\\ [h, e]&=&4g_{41}+4g_{32}+4g_{15}+4g_{11}\\ [h, f]&=&-2g_{-11}-2g_{-15}-2g_{-32}-2g_{-41}\end{array}

Centralizer type:

B^{3}_2+A_1

Killing form square of Cartan element dual to ambient long root: 32
Basis of the centralizer (dimension: 13):

g_{-7}

h_{5}+h_{3}-h_{1}

h_{6}-h_{3}

h_{7}

g_{1}+g_{-29}

g_{3}+g_{-19}

g_{7}

g_{8}-g_{-5}-g_{-9}

g_{9}+g_{5}-g_{-8}

g_{14}+g_{-2}+g_{-24}

g_{19}+g_{-3}

g_{24}+g_{2}+g_{-14}

g_{29}+g_{-1}

Basis of centralizer intersected with cartan (dimension: 3):

h_{6}+h_{5}-h_{1}

-h_{7}

h_{6}-h_{3}

Cartan of centralizer (dimension: 3):

h_{6}+h_{5}-h_{1}

-h_{7}

h_{6}-h_{3}

Cartan-generating semisimple element:

-5h_{7}-2h_{6}+h_{5}+3h_{3}-h_{1}

adjoint action:

\begin{pmatrix}8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -8 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -3 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}

Characteristic polynomial ad H:

x^{13}-79x^{11}+1023x^9-4117x^7+5476x^5-2304x^3

Factorization of characteristic polynomial of ad H: (x )(x )(x )(x -8)(x -3)(x -2)(x -1)(x -1)(x +1)(x +1)(x +2)(x +3)(x +8)
Eigenvalues of ad H:

0

8

3

2

1

-1

-2

-3

-8

13 eigenvectors of ad H:

(0,1,0,0,0,0,0,0,0,0,0,0,0)

(0,0,1,0,0,0,0,0,0,0,0,0,0)

(0,0,0,1,0,0,0,0,0,0,0,0,0)

(1,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,1,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,1,0,0,0,0)

(0,0,0,0,0,0,0,0,0,1,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,1)

(0,0,0,0,1,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,1,0)

(0,0,0,0,0,0,0,1,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,1,0,0)

(0,0,0,0,0,0,1,0,0,0,0,0,0)

Centralizer type: B^{3}_2+A^{1}_1
Reductive components (2 total):
Scalar product computed:

\begin{pmatrix}1/8\\ \end{pmatrix}

Simple basis of Cartan of centralizer (1 total):

-h_{7}

matching e:

g_{-7}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

Linear space basis of intersection of centralizer and ambient Cartan:

-h_{7}

matching e:

g_{-7}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form:

\begin{pmatrix}32\\ \end{pmatrix}

Scalar product computed:

\begin{pmatrix}1/48 & -1/48\\ -1/48 & 1/24\\ \end{pmatrix}

Simple basis of Cartan of centralizer (2 total):

-2h_{7}-4h_{6}-2h_{5}+2h_{3}+2h_{1}

matching e:

g_{14}+g_{-2}+g_{-24}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2\\ \end{pmatrix}

h_{7}+2h_{6}+2h_{5}-2h_{1}

matching e:

g_{29}+g_{-1}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}

Linear space basis of intersection of centralizer and ambient Cartan:

-2h_{7}-4h_{6}-2h_{5}+2h_{3}+2h_{1}

matching e:

g_{14}+g_{-2}+g_{-24}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2\\ \end{pmatrix}

h_{7}+2h_{6}+2h_{5}-2h_{1}

matching e:

g_{29}+g_{-1}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}

Elements in Cartan dual to root system: (1, 1), (-1, -1), (1, 2), (-1, -2), (1, 0), (-1, 0), (0, 1), (0, -1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form:

\begin{pmatrix}192 & -96\\ -96 & 96\\ \end{pmatrix}

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&8h_{7}+16h_{6}+16h_{5}+16h_{4}+12h_{3}+8h_{2}+4h_{1}\\ e&=&x_{1} g_{41}+x_{3} g_{32}+x_{2} g_{15}+x_{4} g_{11}\\ f&=&x_{8} g_{-11}+x_{6} g_{-15}+x_{7} g_{-32}+x_{5} g_{-41}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(2x_{1} x_{5} +2x_{3} x_{7} -8\right)h_{7}+\left(4x_{1} x_{5} +4x_{3} x_{7} -16\right)h_{6}+\left(4x_{1} x_{5} +2x_{3} x_{7} +2x_{4} x_{8} -16\right)h_{5}+\left(2x_{1} x_{5} +2x_{2} x_{6} +2x_{3} x_{7} +2x_{4} x_{8} -16\right)h_{4}+\left(2x_{1} x_{5} +2x_{2} x_{6} +2x_{3} x_{7} -12\right)h_{3}+\left(2x_{1} x_{5} +2x_{2} x_{6} -8\right)h_{2}+\left(2x_{1} x_{5} -4\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{5} -4&=&0\\2x_{1} x_{5} +2x_{2} x_{6} -8&=&0\\2x_{1} x_{5} +2x_{2} x_{6} +2x_{3} x_{7} -12&=&0\\2x_{1} x_{5} +2x_{2} x_{6} +2x_{3} x_{7} +2x_{4} x_{8} -16&=&0\\4x_{1} x_{5} +2x_{3} x_{7} +2x_{4} x_{8} -16&=&0\\4x_{1} x_{5} +4x_{3} x_{7} -16&=&0\\2x_{1} x_{5} +2x_{3} x_{7} -8&=&0\\\end{array}

\begin{array}{rcl}h&=&8h_{7}+16h_{6}+16h_{5}+16h_{4}+12h_{3}+8h_{2}+4h_{1}\\e&=&x_{1} g_{41}+x_{3} g_{32}+x_{2} g_{15}+x_{4} g_{11}\\f&=&g_{-11}+g_{-15}+g_{-32}+g_{-41}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0 & 0\\ 2 & 2 & 0 & 0\\ 2 & 2 & 2 & 0\\ 2 & 2 & 2 & 2\\ 4 & 0 & 2 & 2\\ 4 & 0 & 4 & 0\\ 2 & 0 & 2 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}4\\ 8\\ 12\\ 16\\ 16\\ 16\\ 8\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&8h_{7}+16h_{6}+16h_{5}+16h_{4}+12h_{3}+8h_{2}+4h_{1}\\ e&=&x_{1} g_{41}+x_{3} g_{32}+x_{2} g_{15}+x_{4} g_{11}\\ f&=&x_{8} g_{-11}+x_{6} g_{-15}+x_{7} g_{-32}+x_{5} g_{-41}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{5} -4&=&0\\2x_{1} x_{5} +2x_{2} x_{6} -8&=&0\\2x_{1} x_{5} +2x_{2} x_{6} +2x_{3} x_{7} -12&=&0\\2x_{1} x_{5} +2x_{2} x_{6} +2x_{3} x_{7} +2x_{4} x_{8} -16&=&0\\4x_{1} x_{5} +2x_{3} x_{7} +2x_{4} x_{8} -16&=&0\\4x_{1} x_{5} +4x_{3} x_{7} -16&=&0\\2x_{1} x_{5} +2x_{3} x_{7} -8&=&0\\\end{array}

A^{15}_1

h-characteristic: (2, 0, 0, 0, 0, 0, 2)
Length of the weight dual to h: 30
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 6
Containing regular semisimple subalgebra number 1:

B^{1}_2+2A^{2}_1+A^{1}_1

Containing regular semisimple subalgebra number 2:

C^{1}_3+4A^{1}_1

Containing regular semisimple subalgebra number 3:

B^{1}_2+5A^{1}_1

Containing regular semisimple subalgebra number 4:

C^{1}_3+A^{2}_1+2A^{1}_1

Containing regular semisimple subalgebra number 5:

B^{1}_2+A^{2}_1+3A^{1}_1

Containing regular semisimple subalgebra number 6:

C^{1}_3+2A^{2}_1

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

V_{6\psi}+5V_{4\psi}+21V_{2\psi}+10V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&9h_{7}+16h_{6}+14h_{5}+12h_{4}+10h_{3}+8h_{2}+6h_{1}\\ e&=&4g_{47}+g_{40}+g_{24}+g_{7}+3g_{1}\\ f&=&g_{-1}+g_{-7}+g_{-24}+g_{-40}+g_{-47}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&9h_{7}+16h_{6}+14h_{5}+12h_{4}+10h_{3}+8h_{2}+6h_{1}\\ [h, e]&=&8g_{47}+2g_{40}+2g_{24}+2g_{7}+6g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-7}-2g_{-24}-2g_{-40}-2g_{-47}\end{array}

Centralizer type:

B^{4}_2

Killing form square of Cartan element dual to ambient long root: 32
Basis of the centralizer (dimension: 10):

h_{3}

h_{5}

g_{4}-g_{-16}

g_{6}-g_{-12}

g_{10}-g_{-11}

g_{11}-g_{-10}

g_{12}-g_{-6}

g_{16}-g_{-4}

g_{17}-g_{-22}

g_{22}-g_{-17}

Basis of centralizer intersected with cartan (dimension: 2):

-h_{5}

-h_{3}

Cartan of centralizer (dimension: 2):

-h_{5}

-h_{3}

Cartan-generating semisimple element:

-h_{5}-5h_{3}

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1/2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -1/2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -3 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 5/2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -5/2\\ \end{pmatrix}

Characteristic polynomial ad H:

x^{10}-39/2x^8+1953/16x^6-4069/16x^4+225/4x^2

Factorization of characteristic polynomial of ad H: 1/16(x )(x )(x -3)(x -2)(x +2)(x +3)(2x -5)(2x -1)(2x +1)(2x +5)
Eigenvalues of ad H:

0

3

2

-2

-3

5/2

1/2

-1/2

-5/2

10 eigenvectors of ad H:

(1,0,0,0,0,0,0,0,0,0)

(0,1,0,0,0,0,0,0,0,0)

(0,0,1,0,0,0,0,0,0,0)

(0,0,0,0,0,1,0,0,0,0)

(0,0,0,0,1,0,0,0,0,0)

(0,0,0,0,0,0,0,1,0,0)

(0,0,0,0,0,0,0,0,1,0)

(0,0,0,1,0,0,0,0,0,0)

(0,0,0,0,0,0,1,0,0,0)

(0,0,0,0,0,0,0,0,0,1)

Centralizer type: B^{4}_2
Reductive components (1 total):
Scalar product computed:

\begin{pmatrix}1/32 & -1/64\\ -1/64 & 1/64\\ \end{pmatrix}

Simple basis of Cartan of centralizer (2 total):

2h_{5}-2h_{3}

matching e:

g_{11}-g_{-10}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}

-4h_{5}

matching e:

g_{6}-g_{-12}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

Linear space basis of intersection of centralizer and ambient Cartan:

2h_{5}-2h_{3}

matching e:

g_{11}-g_{-10}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}

-4h_{5}

matching e:

g_{6}-g_{-12}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

Elements in Cartan dual to root system: (1, 1), (-1, -1), (1, 0), (-1, 0), (2, 1), (-2, -1), (0, 1), (0, -1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form:

\begin{pmatrix}128 & -128\\ -128 & 256\\ \end{pmatrix}

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&9h_{7}+16h_{6}+14h_{5}+12h_{4}+10h_{3}+8h_{2}+6h_{1}\\ e&=&x_{1} g_{47}+x_{3} g_{40}+x_{4} g_{24}+x_{5} g_{7}+x_{2} g_{1}\\ f&=&x_{7} g_{-1}+x_{10} g_{-7}+x_{9} g_{-24}+x_{8} g_{-40}+x_{6} g_{-47}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(x_{1} x_{6} +2x_{3} x_{8} +2x_{4} x_{9} +x_{5} x_{10} -9\right)h_{7}+\left(2x_{1} x_{6} +4x_{3} x_{8} +4x_{4} x_{9} -16\right)h_{6}+\left(2x_{1} x_{6} +4x_{3} x_{8} +2x_{4} x_{9} -14\right)h_{5}+\left(2x_{1} x_{6} +4x_{3} x_{8} -12\right)h_{4}+\left(2x_{1} x_{6} +2x_{3} x_{8} -10\right)h_{3}+\left(2x_{1} x_{6} -8\right)h_{2}+\left(2x_{2} x_{7} -6\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{6} -8&=&0\\2x_{1} x_{6} +2x_{3} x_{8} -10&=&0\\2x_{1} x_{6} +4x_{3} x_{8} -12&=&0\\2x_{1} x_{6} +4x_{3} x_{8} +2x_{4} x_{9} -14&=&0\\2x_{1} x_{6} +4x_{3} x_{8} +4x_{4} x_{9} -16&=&0\\x_{1} x_{6} +2x_{3} x_{8} +2x_{4} x_{9} +x_{5} x_{10} -9&=&0\\2x_{2} x_{7} -6&=&0\\\end{array}

\begin{array}{rcl}h&=&9h_{7}+16h_{6}+14h_{5}+12h_{4}+10h_{3}+8h_{2}+6h_{1}\\e&=&x_{1} g_{47}+x_{3} g_{40}+x_{4} g_{24}+x_{5} g_{7}+x_{2} g_{1}\\f&=&g_{-1}+g_{-7}+g_{-24}+g_{-40}+g_{-47}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0 & 0 & 0\\ 2 & 0 & 2 & 0 & 0\\ 2 & 0 & 4 & 0 & 0\\ 2 & 0 & 4 & 2 & 0\\ 2 & 0 & 4 & 4 & 0\\ 1 & 0 & 2 & 2 & 1\\ 0 & 2 & 0 & 0 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}8\\ 10\\ 12\\ 14\\ 16\\ 9\\ 6\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&9h_{7}+16h_{6}+14h_{5}+12h_{4}+10h_{3}+8h_{2}+6h_{1}\\ e&=&x_{1} g_{47}+x_{3} g_{40}+x_{4} g_{24}+x_{5} g_{7}+x_{2} g_{1}\\ f&=&x_{7} g_{-1}+x_{10} g_{-7}+x_{9} g_{-24}+x_{8} g_{-40}+x_{6} g_{-47}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{6} -8&=&0\\2x_{1} x_{6} +2x_{3} x_{8} -10&=&0\\2x_{1} x_{6} +4x_{3} x_{8} -12&=&0\\2x_{1} x_{6} +4x_{3} x_{8} +2x_{4} x_{9} -14&=&0\\2x_{1} x_{6} +4x_{3} x_{8} +4x_{4} x_{9} -16&=&0\\x_{1} x_{6} +2x_{3} x_{8} +2x_{4} x_{9} +x_{5} x_{10} -9&=&0\\2x_{2} x_{7} -6&=&0\\\end{array}

A^{14}_1

h-characteristic: (2, 0, 0, 0, 0, 1, 0)
Length of the weight dual to h: 28
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 5
Containing regular semisimple subalgebra number 1:

B^{1}_2+4A^{1}_1

Containing regular semisimple subalgebra number 2:

B^{1}_2+A^{2}_1+2A^{1}_1

Containing regular semisimple subalgebra number 3:

B^{1}_2+2A^{2}_1

Containing regular semisimple subalgebra number 4:

C^{1}_3+3A^{1}_1

Containing regular semisimple subalgebra number 5:

C^{1}_3+A^{2}_1+A^{1}_1

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

V_{6\psi}+4V_{4\psi}+2V_{3\psi}+15V_{2\psi}+8V_{\psi}+9V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&8h_{7}+16h_{6}+14h_{5}+12h_{4}+10h_{3}+8h_{2}+6h_{1}\\ e&=&4g_{47}+g_{43}+g_{37}+g_{29}+g_{19}+3g_{1}\\ f&=&g_{-1}+g_{-19}+g_{-29}+g_{-37}+g_{-43}+g_{-47}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&8h_{7}+16h_{6}+14h_{5}+12h_{4}+10h_{3}+8h_{2}+6h_{1}\\ [h, e]&=&8g_{47}+2g_{43}+2g_{37}+2g_{29}+2g_{19}+6g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-19}-2g_{-29}-2g_{-37}-2g_{-43}-2g_{-47}\end{array}

Centralizer type:

2A^{4}_1+A_1

Killing form square of Cartan element dual to ambient long root: 32
Basis of the centralizer (dimension: 9):

g_{-7}

h_{7}

g_{3}-g_{-3}

g_{4}-g_{-4}

g_{5}-g_{-5}

g_{7}

g_{10}-g_{-10}

g_{11}-g_{-11}

g_{16}-g_{-16}

Basis of centralizer intersected with cartan (dimension: 1):

-h_{7}

Cartan of centralizer (dimension: 3):

-g_{7}-h_{7}-g_{-7}

g_{11}+g_{10}-g_{-10}-g_{-11}

g_{16}+g_{5}+g_{4}+g_{3}-g_{-3}-g_{-4}-g_{-5}-g_{-16}

Cartan-generating semisimple element:

g_{16}+g_{11}+g_{10}+g_{7}+g_{5}+g_{4}+g_{3}+h_{7}-g_{-3}-g_{-4}-g_{-5}+g_{-7}-g_{-10}-g_{-11}-g_{-16}

adjoint action:

\begin{pmatrix}-2 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 1 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 1 & -1 & 1\\ 0 & 0 & 1 & 0 & -1 & 0 & -1 & 1 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 1 & -1 & -1\\ 0 & -2 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & -1 & 1 & -1 & 0 & 0 & 0 & 1\\ 0 & 0 & 1 & -1 & 1 & 0 & 0 & 0 & -1\\ 0 & 0 & -1 & 0 & 1 & 0 & -1 & 1 & 0\\ \end{pmatrix}

Characteristic polynomial ad H:

x^9+4x^7-64x^5-256x^3

Factorization of characteristic polynomial of ad H: (x )(x )(x )(x^2-8)(x^2+4)(x^2+8)
Eigenvalues of ad H:

0

2\sqrt{2}

-2\sqrt{2}

2\sqrt{-1}

-2\sqrt{-1}

2\sqrt{-2}

-2\sqrt{-2}

9 eigenvectors of ad H:

(1,1,0,0,0,1,0,0,0)

(0,0,0,0,0,0,1,1,0)

(0,0,1,1,1,0,0,0,1)

(22−3,−2+1,0,0,0,1,0,0,0)

(−22−3,2+1,0,0,0,1,0,0,0)

(0,0,−−1,−1,−1,0,0,0,1)

(0,0,−1,−1,−−1,0,0,0,1)

(0,0,−1,1,−1,0,−−2,−2,1)

(0,0,−1,1,−1,0,−2,−−2,1)

Centralizer type: 2A^{4}_1+A^{1}_1
Reductive components (3 total):
Scalar product computed:

\begin{pmatrix}1/8\\ \end{pmatrix}

Simple basis of Cartan of centralizer (1 total):

1/2\sqrt{2}g_{7}+1/2\sqrt{2}h_{7}+1/2\sqrt{2}g_{-7}

matching e:

g_{7}+\left(-\sqrt{2}+1\right)h_{7}+\left(2\sqrt{2}-3\right)g_{-7}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}-\sqrt{2} & \sqrt{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 1/2\sqrt{2} & 0 & 0 & 0 & 0 & -1/2\sqrt{2} & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -\sqrt{2} & 0 & 0 & 0 & \sqrt{2} & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

Linear space basis of intersection of centralizer and ambient Cartan:

1/2\sqrt{2}g_{7}+1/2\sqrt{2}h_{7}+1/2\sqrt{2}g_{-7}

matching e:

g_{7}+\left(-\sqrt{2}+1\right)h_{7}+\left(2\sqrt{2}-3\right)g_{-7}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}-\sqrt{2} & \sqrt{2} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 1/2\sqrt{2} & 0 & 0 & 0 & 0 & -1/2\sqrt{2} & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -\sqrt{2} & 0 & 0 & 0 & \sqrt{2} & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form:

\begin{pmatrix}32\\ \end{pmatrix}

Scalar product computed:

\begin{pmatrix}1/32\\ \end{pmatrix}

Simple basis of Cartan of centralizer (1 total):

-\sqrt{-1}g_{11}-\sqrt{-1}g_{10}+\sqrt{-1}g_{-10}+\sqrt{-1}g_{-11}

matching e:

g_{16}+\sqrt{-1}g_{5}-g_{4}-\sqrt{-1}g_{3}+\sqrt{-1}g_{-3}+g_{-4}-\sqrt{-1}g_{-5}-g_{-16}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & \sqrt{-1} & 0 & 0 & 0 & 0 & -\sqrt{-1}\\ 0 & 0 & -\sqrt{-1} & 0 & \sqrt{-1} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -\sqrt{-1} & 0 & 0 & 0 & 0 & \sqrt{-1}\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & \sqrt{-1} & 0 & -\sqrt{-1} & 0 & 0 & 0 & 0\\ \end{pmatrix}

Linear space basis of intersection of centralizer and ambient Cartan:

-\sqrt{-1}g_{11}-\sqrt{-1}g_{10}+\sqrt{-1}g_{-10}+\sqrt{-1}g_{-11}

matching e:

g_{16}+\sqrt{-1}g_{5}-g_{4}-\sqrt{-1}g_{3}+\sqrt{-1}g_{-3}+g_{-4}-\sqrt{-1}g_{-5}-g_{-16}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & \sqrt{-1} & 0 & 0 & 0 & 0 & -\sqrt{-1}\\ 0 & 0 & -\sqrt{-1} & 0 & \sqrt{-1} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -\sqrt{-1} & 0 & 0 & 0 & 0 & \sqrt{-1}\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & \sqrt{-1} & 0 & -\sqrt{-1} & 0 & 0 & 0 & 0\\ \end{pmatrix}

Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form:

\begin{pmatrix}128\\ \end{pmatrix}

Scalar product computed:

\begin{pmatrix}1/32\\ \end{pmatrix}

Simple basis of Cartan of centralizer (1 total):

-1/2\sqrt{-2}g_{16}-1/2\sqrt{-2}g_{5}-1/2\sqrt{-2}g_{4}-1/2\sqrt{-2}g_{3}+1/2\sqrt{-2}g_{-3}+1/2\sqrt{-2}g_{-4}+1/2\sqrt{-2}g_{-5}+1/2\sqrt{-2}g_{-16}

matching e:

g_{16}+\sqrt{-2}g_{11}-\sqrt{-2}g_{10}-g_{5}+g_{4}-g_{3}+g_{-3}-g_{-4}+g_{-5}+\sqrt{-2}g_{-10}-\sqrt{-2}g_{-11}-g_{-16}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -1/2\sqrt{-2} & 1/2\sqrt{-2} & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -1/2\sqrt{-2} & 1/2\sqrt{-2} & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 1/2\sqrt{-2} & 0 & 0 & 0 & -1/2\sqrt{-2}\\ 0 & 0 & -1/2\sqrt{-2} & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 0 & 0 & 0 & 1/2\sqrt{-2}\\ 0 & 0 & 0 & 0 & 0 & 0 & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 0\\ \end{pmatrix}

Linear space basis of intersection of centralizer and ambient Cartan:

-1/2\sqrt{-2}g_{16}-1/2\sqrt{-2}g_{5}-1/2\sqrt{-2}g_{4}-1/2\sqrt{-2}g_{3}+1/2\sqrt{-2}g_{-3}+1/2\sqrt{-2}g_{-4}+1/2\sqrt{-2}g_{-5}+1/2\sqrt{-2}g_{-16}

matching e:

g_{16}+\sqrt{-2}g_{11}-\sqrt{-2}g_{10}-g_{5}+g_{4}-g_{3}+g_{-3}-g_{-4}+g_{-5}+\sqrt{-2}g_{-10}-\sqrt{-2}g_{-11}-g_{-16}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -1/2\sqrt{-2} & 1/2\sqrt{-2} & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -1/2\sqrt{-2} & 1/2\sqrt{-2} & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 1/2\sqrt{-2} & 0 & 0 & 0 & -1/2\sqrt{-2}\\ 0 & 0 & -1/2\sqrt{-2} & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 0 & 0 & 0 & 1/2\sqrt{-2}\\ 0 & 0 & 0 & 0 & 0 & 0 & 1/2\sqrt{-2} & -1/2\sqrt{-2} & 0\\ \end{pmatrix}

Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form:

\begin{pmatrix}128\\ \end{pmatrix}

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&8h_{7}+16h_{6}+14h_{5}+12h_{4}+10h_{3}+8h_{2}+6h_{1}\\ e&=&x_{1} g_{47}+x_{3} g_{43}+x_{4} g_{37}+x_{5} g_{29}+x_{6} g_{19}+x_{2} g_{1}\\ f&=&x_{8} g_{-1}+x_{12} g_{-19}+x_{11} g_{-29}+x_{10} g_{-37}+x_{9} g_{-43}+x_{7} g_{-47}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(x_{1} x_{7} +x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -8\right)h_{7}+\left(2x_{1} x_{7} +2x_{3} x_{9} +2x_{4} x_{10} +2x_{5} x_{11} +2x_{6} x_{12} -16\right)h_{6}+\left(2x_{1} x_{7} +2x_{3} x_{9} +2x_{4} x_{10} +2x_{5} x_{11} -14\right)h_{5}+\left(2x_{1} x_{7} +2x_{3} x_{9} +2x_{4} x_{10} -12\right)h_{4}+\left(2x_{1} x_{7} +2x_{3} x_{9} -10\right)h_{3}+\left(2x_{1} x_{7} -8\right)h_{2}+\left(2x_{2} x_{8} -6\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{7} -8&=&0\\2x_{1} x_{7} +2x_{3} x_{9} -10&=&0\\2x_{1} x_{7} +2x_{3} x_{9} +2x_{4} x_{10} -12&=&0\\2x_{1} x_{7} +2x_{3} x_{9} +2x_{4} x_{10} +2x_{5} x_{11} -14&=&0\\2x_{1} x_{7} +2x_{3} x_{9} +2x_{4} x_{10} +2x_{5} x_{11} +2x_{6} x_{12} -16&=&0\\x_{1} x_{7} +x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -8&=&0\\2x_{2} x_{8} -6&=&0\\\end{array}

\begin{array}{rcl}h&=&8h_{7}+16h_{6}+14h_{5}+12h_{4}+10h_{3}+8h_{2}+6h_{1}\\e&=&x_{1} g_{47}+x_{3} g_{43}+x_{4} g_{37}+x_{5} g_{29}+x_{6} g_{19}+x_{2} g_{1}\\f&=&g_{-1}+g_{-19}+g_{-29}+g_{-37}+g_{-43}+g_{-47}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0\\ 2 & 0 & 2 & 0 & 0 & 0\\ 2 & 0 & 2 & 2 & 0 & 0\\ 2 & 0 & 2 & 2 & 2 & 0\\ 2 & 0 & 2 & 2 & 2 & 2\\ 1 & 0 & 1 & 1 & 1 & 1\\ 0 & 2 & 0 & 0 & 0 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}8\\ 10\\ 12\\ 14\\ 16\\ 8\\ 6\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&8h_{7}+16h_{6}+14h_{5}+12h_{4}+10h_{3}+8h_{2}+6h_{1}\\ e&=&x_{1} g_{47}+x_{3} g_{43}+x_{4} g_{37}+x_{5} g_{29}+x_{6} g_{19}+x_{2} g_{1}\\ f&=&x_{8} g_{-1}+x_{12} g_{-19}+x_{11} g_{-29}+x_{10} g_{-37}+x_{9} g_{-43}+x_{7} g_{-47}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{7} -8&=&0\\2x_{1} x_{7} +2x_{3} x_{9} -10&=&0\\2x_{1} x_{7} +2x_{3} x_{9} +2x_{4} x_{10} -12&=&0\\2x_{1} x_{7} +2x_{3} x_{9} +2x_{4} x_{10} +2x_{5} x_{11} -14&=&0\\2x_{1} x_{7} +2x_{3} x_{9} +2x_{4} x_{10} +2x_{5} x_{11} +2x_{6} x_{12} -16&=&0\\x_{1} x_{7} +x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -8&=&0\\2x_{2} x_{8} -6&=&0\\\end{array}

A^{13}_1

h-characteristic: (2, 0, 0, 0, 1, 0, 0)
Length of the weight dual to h: 26
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 4
Containing regular semisimple subalgebra number 1:

B^{1}_2+3A^{1}_1

Containing regular semisimple subalgebra number 2:

B^{1}_2+A^{2}_1+A^{1}_1

Containing regular semisimple subalgebra number 3:

C^{1}_3+2A^{1}_1

Containing regular semisimple subalgebra number 4:

C^{1}_3+A^{2}_1

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

V_{6\psi}+3V_{4\psi}+4V_{3\psi}+10V_{2\psi}+12V_{\psi}+13V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&7h_{7}+14h_{6}+14h_{5}+12h_{4}+10h_{3}+8h_{2}+6h_{1}\\ e&=&4g_{47}+g_{43}+g_{37}+g_{29}+3g_{1}\\ f&=&g_{-1}+g_{-29}+g_{-37}+g_{-43}+g_{-47}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&7h_{7}+14h_{6}+14h_{5}+12h_{4}+10h_{3}+8h_{2}+6h_{1}\\ [h, e]&=&8g_{47}+2g_{43}+2g_{37}+2g_{29}+6g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-29}-2g_{-37}-2g_{-43}-2g_{-47}\end{array}

Centralizer type:

B_2+A^{8}_1

Killing form square of Cartan element dual to ambient long root: 32
Basis of the centralizer (dimension: 13):

g_{-19}

g_{-13}

g_{-7}

g_{-6}

h_{6}

h_{7}

g_{3}-g_{-3}

g_{4}-g_{-4}

g_{6}

g_{7}

g_{10}-g_{-10}

g_{13}

g_{19}

Basis of centralizer intersected with cartan (dimension: 2):

-h_{7}

-h_{6}

Cartan of centralizer (dimension: 3):

-g_{19}-g_{13}-g_{7}-3h_{7}-3h_{6}-g_{-7}-g_{-13}-g_{-19}

-g_{6}+2h_{7}+2h_{6}-g_{-6}

g_{10}+g_{4}+g_{3}-g_{-3}-g_{-4}-g_{-10}

Cartan-generating semisimple element:

g_{19}+g_{13}+g_{10}+g_{7}+g_{6}+g_{4}+g_{3}+h_{7}+h_{6}-g_{-3}-g_{-4}+g_{-6}+g_{-7}-g_{-10}+g_{-13}+g_{-19}

adjoint action:

\begin{pmatrix}-1 & -2 & 0 & 2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ -1 & -1 & -1 & 1 & 0 & 1 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & -2 & -1 & 0 & -1 & 2 & 0 & 0 & 2 & 0 & 0 & 0 & 0\\ 1 & 1 & 0 & 0 & 1 & -1 & 0 & 0 & 0 & -1 & 0 & -1 & 0\\ 2 & 2 & 0 & 2 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & -2 & -2\\ 1 & 2 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & -2 & -1\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 1 & 1 & 0 & -1 & 1 & 0 & 0 & 0 & 0 & 0 & -1 & -1\\ 0 & 0 & 0 & -2 & 1 & -2 & 0 & 0 & 0 & 1 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & -1 & 0 & 0 & -1 & 1 & 0 & 1 & 1\\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & -2 & 0 & 0 & 2 & 1\\ \end{pmatrix}

Characteristic polynomial ad H:

x^{13}-36x^{11}+282x^9-64x^7-2883x^5+2700x^3

Factorization of characteristic polynomial of ad H: (x )(x )(x )(x -5)(x -3)(x -2)(x -1)(x +1)(x +2)(x +3)(x +5)(x^2+3)
Eigenvalues of ad H:

0

5

3

2

1

-1

-2

-3

-5

\sqrt{-3}

-\sqrt{-3}

13 eigenvectors of ad H:

(0,0,0,1,−2,−2,0,0,1,0,0,0,0)

(0,0,0,0,0,0,1,1,0,0,1,0,0)

(1,1,1,0,3,3,0,0,0,1,0,1,1)

(−14,−14,−14,−12,−1,−1,0,0,−12,1,0,1,1)

(−1,0,1,−1,−2,0,0,0,1,0,0,0,0)

(0,0,0,14,−12,0,0,0,−14,−1,0,0,1)

(1,−1,1,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,1,0,−1,1)

(4,0,−4,−1,−2,0,0,0,1,0,0,0,0)

(0,0,0,−1,2,0,0,0,1,−1,0,0,1)

(−4,−4,−4,2,4,4,0,0,2,1,0,1,1)

(0,0,0,0,0,0,−12−3−12,12−3−12,0,0,1,0,0)

(0,0,0,0,0,0,12−3−12,−12−3−12,0,0,1,0,0)

Centralizer type: B^{1}_2+A^{8}_1
Reductive components (2 total):
Scalar product computed:

\begin{pmatrix}1/16 & -1/16\\ -1/16 & 1/8\\ \end{pmatrix}

Simple basis of Cartan of centralizer (2 total):

2/5g_{19}+2/5g_{13}+2/5g_{7}-1/5g_{6}+8/5h_{7}+8/5h_{6}-1/5g_{-6}+2/5g_{-7}+2/5g_{-13}+2/5g_{-19}

matching e:

g_{19}-g_{7}-1/4g_{6}-1/2h_{6}+1/4g_{-6}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}-8/5 & 2/5 & 0 & 4/5 & 2/5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 1/5 & -8/5 & 1/5 & 2/5 & 0 & 2/5 & 0 & 0 & 2/5 & 0 & 0 & 0 & 0\\ 0 & 2/5 & -8/5 & 0 & -2/5 & 4/5 & 0 & 0 & 4/5 & 0 & 0 & 0 & 0\\ 2/5 & 2/5 & 0 & 0 & -1/5 & 1/5 & 0 & 0 & 0 & -2/5 & 0 & -2/5 & 0\\ 4/5 & 4/5 & 0 & -2/5 & 0 & 0 & 0 & 0 & 2/5 & 0 & 0 & -4/5 & -4/5\\ 2/5 & 4/5 & 2/5 & 0 & 0 & 0 & 0 & 0 & 0 & -2/5 & 0 & -4/5 & -2/5\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2/5 & 2/5 & 0 & 1/5 & -1/5 & 0 & 0 & 0 & 0 & 0 & -2/5 & -2/5\\ 0 & 0 & 0 & -4/5 & 2/5 & -4/5 & 0 & 0 & 0 & 8/5 & 0 & -2/5 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2/5 & 0 & -2/5 & 0 & 0 & -2/5 & -1/5 & 0 & 8/5 & -1/5\\ 0 & 0 & 0 & 0 & -2/5 & 0 & 0 & 0 & -4/5 & 0 & 0 & -2/5 & 8/5\\ \end{pmatrix}

1/2g_{6}-h_{7}-h_{6}+1/2g_{-6}

matching e:

g_{-7}-g_{-13}+g_{-19}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ -1/2 & 1 & -1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1/2 & -1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1/2 & 1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1/2 & 0 & -1 & 1/2\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & -1\\ \end{pmatrix}

Linear space basis of intersection of centralizer and ambient Cartan:

2/5g_{19}+2/5g_{13}+2/5g_{7}-1/5g_{6}+8/5h_{7}+8/5h_{6}-1/5g_{-6}+2/5g_{-7}+2/5g_{-13}+2/5g_{-19}

matching e:

g_{19}-g_{7}-1/4g_{6}-1/2h_{6}+1/4g_{-6}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}-8/5 & 2/5 & 0 & 4/5 & 2/5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 1/5 & -8/5 & 1/5 & 2/5 & 0 & 2/5 & 0 & 0 & 2/5 & 0 & 0 & 0 & 0\\ 0 & 2/5 & -8/5 & 0 & -2/5 & 4/5 & 0 & 0 & 4/5 & 0 & 0 & 0 & 0\\ 2/5 & 2/5 & 0 & 0 & -1/5 & 1/5 & 0 & 0 & 0 & -2/5 & 0 & -2/5 & 0\\ 4/5 & 4/5 & 0 & -2/5 & 0 & 0 & 0 & 0 & 2/5 & 0 & 0 & -4/5 & -4/5\\ 2/5 & 4/5 & 2/5 & 0 & 0 & 0 & 0 & 0 & 0 & -2/5 & 0 & -4/5 & -2/5\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2/5 & 2/5 & 0 & 1/5 & -1/5 & 0 & 0 & 0 & 0 & 0 & -2/5 & -2/5\\ 0 & 0 & 0 & -4/5 & 2/5 & -4/5 & 0 & 0 & 0 & 8/5 & 0 & -2/5 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -2/5 & 0 & -2/5 & 0 & 0 & -2/5 & -1/5 & 0 & 8/5 & -1/5\\ 0 & 0 & 0 & 0 & -2/5 & 0 & 0 & 0 & -4/5 & 0 & 0 & -2/5 & 8/5\\ \end{pmatrix}

1/2g_{6}-h_{7}-h_{6}+1/2g_{-6}

matching e:

g_{-7}-g_{-13}+g_{-19}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}1 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ -1/2 & 1 & -1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1/2 & -1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1/2 & 1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1/2 & 0 & -1 & 1/2\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & -1\\ \end{pmatrix}

Elements in Cartan dual to root system: (1, 1), (-1, -1), (1, 2), (-1, -2), (1, 0), (-1, 0), (0, 1), (0, -1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form:

\begin{pmatrix}64 & -32\\ -32 & 32\\ \end{pmatrix}

Scalar product computed:

\begin{pmatrix}1/64\\ \end{pmatrix}

Simple basis of Cartan of centralizer (1 total):

-2/3\sqrt{-3}g_{10}-2/3\sqrt{-3}g_{4}-2/3\sqrt{-3}g_{3}+2/3\sqrt{-3}g_{-3}+2/3\sqrt{-3}g_{-4}+2/3\sqrt{-3}g_{-10}

matching e:

g_{10}+\left(1/2\sqrt{-3}-1/2\right)g_{4}+\left(-1/2\sqrt{-3}-1/2\right)g_{3}+\left(1/2\sqrt{-3}+1/2\right)g_{-3}+\left(-1/2\sqrt{-3}+1/2\right)g_{-4}-g_{-10}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2/3\sqrt{-3} & 0 & 0 & -2/3\sqrt{-3} & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -2/3\sqrt{-3} & 0 & 0 & 0 & 2/3\sqrt{-3} & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2/3\sqrt{-3} & -2/3\sqrt{-3} & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

Linear space basis of intersection of centralizer and ambient Cartan:

-2/3\sqrt{-3}g_{10}-2/3\sqrt{-3}g_{4}-2/3\sqrt{-3}g_{3}+2/3\sqrt{-3}g_{-3}+2/3\sqrt{-3}g_{-4}+2/3\sqrt{-3}g_{-10}

matching e:

g_{10}+\left(1/2\sqrt{-3}-1/2\right)g_{4}+\left(-1/2\sqrt{-3}-1/2\right)g_{3}+\left(1/2\sqrt{-3}+1/2\right)g_{-3}+\left(-1/2\sqrt{-3}+1/2\right)g_{-4}-g_{-10}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2/3\sqrt{-3} & 0 & 0 & -2/3\sqrt{-3} & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -2/3\sqrt{-3} & 0 & 0 & 0 & 2/3\sqrt{-3} & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2/3\sqrt{-3} & -2/3\sqrt{-3} & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form:

\begin{pmatrix}256\\ \end{pmatrix}

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&7h_{7}+14h_{6}+14h_{5}+12h_{4}+10h_{3}+8h_{2}+6h_{1}\\ e&=&x_{1} g_{47}+x_{3} g_{43}+x_{4} g_{37}+x_{5} g_{29}+x_{2} g_{1}\\ f&=&x_{7} g_{-1}+x_{10} g_{-29}+x_{9} g_{-37}+x_{8} g_{-43}+x_{6} g_{-47}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(x_{1} x_{6} +x_{3} x_{8} +x_{4} x_{9} +x_{5} x_{10} -7\right)h_{7}+\left(2x_{1} x_{6} +2x_{3} x_{8} +2x_{4} x_{9} +2x_{5} x_{10} -14\right)h_{6}+\left(2x_{1} x_{6} +2x_{3} x_{8} +2x_{4} x_{9} +2x_{5} x_{10} -14\right)h_{5}+\left(2x_{1} x_{6} +2x_{3} x_{8} +2x_{4} x_{9} -12\right)h_{4}+\left(2x_{1} x_{6} +2x_{3} x_{8} -10\right)h_{3}+\left(2x_{1} x_{6} -8\right)h_{2}+\left(2x_{2} x_{7} -6\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{6} -8&=&0\\2x_{1} x_{6} +2x_{3} x_{8} -10&=&0\\2x_{1} x_{6} +2x_{3} x_{8} +2x_{4} x_{9} -12&=&0\\2x_{1} x_{6} +2x_{3} x_{8} +2x_{4} x_{9} +2x_{5} x_{10} -14&=&0\\2x_{1} x_{6} +2x_{3} x_{8} +2x_{4} x_{9} +2x_{5} x_{10} -14&=&0\\x_{1} x_{6} +x_{3} x_{8} +x_{4} x_{9} +x_{5} x_{10} -7&=&0\\2x_{2} x_{7} -6&=&0\\\end{array}

\begin{array}{rcl}h&=&7h_{7}+14h_{6}+14h_{5}+12h_{4}+10h_{3}+8h_{2}+6h_{1}\\e&=&x_{1} g_{47}+x_{3} g_{43}+x_{4} g_{37}+x_{5} g_{29}+x_{2} g_{1}\\f&=&g_{-1}+g_{-29}+g_{-37}+g_{-43}+g_{-47}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0 & 0 & 0\\ 2 & 0 & 2 & 0 & 0\\ 2 & 0 & 2 & 2 & 0\\ 2 & 0 & 2 & 2 & 2\\ 2 & 0 & 2 & 2 & 2\\ 1 & 0 & 1 & 1 & 1\\ 0 & 2 & 0 & 0 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}8\\ 10\\ 12\\ 14\\ 14\\ 7\\ 6\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&7h_{7}+14h_{6}+14h_{5}+12h_{4}+10h_{3}+8h_{2}+6h_{1}\\ e&=&x_{1} g_{47}+x_{3} g_{43}+x_{4} g_{37}+x_{5} g_{29}+x_{2} g_{1}\\ f&=&x_{7} g_{-1}+x_{10} g_{-29}+x_{9} g_{-37}+x_{8} g_{-43}+x_{6} g_{-47}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{6} -8&=&0\\2x_{1} x_{6} +2x_{3} x_{8} -10&=&0\\2x_{1} x_{6} +2x_{3} x_{8} +2x_{4} x_{9} -12&=&0\\2x_{1} x_{6} +2x_{3} x_{8} +2x_{4} x_{9} +2x_{5} x_{10} -14&=&0\\2x_{1} x_{6} +2x_{3} x_{8} +2x_{4} x_{9} +2x_{5} x_{10} -14&=&0\\x_{1} x_{6} +x_{3} x_{8} +x_{4} x_{9} +x_{5} x_{10} -7&=&0\\2x_{2} x_{7} -6&=&0\\\end{array}

A^{12}_1

h-characteristic: (2, 0, 0, 1, 0, 0, 0)
Length of the weight dual to h: 24
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1:

B^{1}_2+2A^{1}_1

Containing regular semisimple subalgebra number 2:

B^{1}_2+A^{2}_1

Containing regular semisimple subalgebra number 3:

C^{1}_3+A^{1}_1

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

V_{6\psi}+2V_{4\psi}+6V_{3\psi}+6V_{2\psi}+12V_{\psi}+22V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&6h_{7}+12h_{6}+12h_{5}+12h_{4}+10h_{3}+8h_{2}+6h_{1}\\ e&=&4g_{47}+g_{43}+g_{37}+3g_{1}\\ f&=&g_{-1}+g_{-37}+g_{-43}+g_{-47}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&6h_{7}+12h_{6}+12h_{5}+12h_{4}+10h_{3}+8h_{2}+6h_{1}\\ [h, e]&=&8g_{47}+2g_{43}+2g_{37}+6g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-37}-2g_{-43}-2g_{-47}\end{array}

Centralizer type:

C_3

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&6h_{7}+12h_{6}+12h_{5}+12h_{4}+10h_{3}+8h_{2}+6h_{1}\\ e&=&x_{1} g_{47}+x_{3} g_{43}+x_{4} g_{37}+x_{2} g_{1}\\ f&=&x_{6} g_{-1}+x_{8} g_{-37}+x_{7} g_{-43}+x_{5} g_{-47}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(x_{1} x_{5} +x_{3} x_{7} +x_{4} x_{8} -6\right)h_{7}+\left(2x_{1} x_{5} +2x_{3} x_{7} +2x_{4} x_{8} -12\right)h_{6}+\left(2x_{1} x_{5} +2x_{3} x_{7} +2x_{4} x_{8} -12\right)h_{5}+\left(2x_{1} x_{5} +2x_{3} x_{7} +2x_{4} x_{8} -12\right)h_{4}+\left(2x_{1} x_{5} +2x_{3} x_{7} -10\right)h_{3}+\left(2x_{1} x_{5} -8\right)h_{2}+\left(2x_{2} x_{6} -6\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{5} -8&=&0\\2x_{1} x_{5} +2x_{3} x_{7} -10&=&0\\2x_{1} x_{5} +2x_{3} x_{7} +2x_{4} x_{8} -12&=&0\\2x_{1} x_{5} +2x_{3} x_{7} +2x_{4} x_{8} -12&=&0\\2x_{1} x_{5} +2x_{3} x_{7} +2x_{4} x_{8} -12&=&0\\x_{1} x_{5} +x_{3} x_{7} +x_{4} x_{8} -6&=&0\\2x_{2} x_{6} -6&=&0\\\end{array}

\begin{array}{rcl}h&=&6h_{7}+12h_{6}+12h_{5}+12h_{4}+10h_{3}+8h_{2}+6h_{1}\\e&=&x_{1} g_{47}+x_{3} g_{43}+x_{4} g_{37}+x_{2} g_{1}\\f&=&g_{-1}+g_{-37}+g_{-43}+g_{-47}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0 & 0\\ 2 & 0 & 2 & 0\\ 2 & 0 & 2 & 2\\ 2 & 0 & 2 & 2\\ 2 & 0 & 2 & 2\\ 1 & 0 & 1 & 1\\ 0 & 2 & 0 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}8\\ 10\\ 12\\ 12\\ 12\\ 6\\ 6\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&6h_{7}+12h_{6}+12h_{5}+12h_{4}+10h_{3}+8h_{2}+6h_{1}\\ e&=&x_{1} g_{47}+x_{3} g_{43}+x_{4} g_{37}+x_{2} g_{1}\\ f&=&x_{6} g_{-1}+x_{8} g_{-37}+x_{7} g_{-43}+x_{5} g_{-47}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{5} -8&=&0\\2x_{1} x_{5} +2x_{3} x_{7} -10&=&0\\2x_{1} x_{5} +2x_{3} x_{7} +2x_{4} x_{8} -12&=&0\\2x_{1} x_{5} +2x_{3} x_{7} +2x_{4} x_{8} -12&=&0\\2x_{1} x_{5} +2x_{3} x_{7} +2x_{4} x_{8} -12&=&0\\x_{1} x_{5} +x_{3} x_{7} +x_{4} x_{8} -6&=&0\\2x_{2} x_{6} -6&=&0\\\end{array}

A^{12}_1

h-characteristic: (0, 1, 0, 0, 0, 1, 0)
Length of the weight dual to h: 24
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1:

A^{2}_2+4A^{1}_1

Containing regular semisimple subalgebra number 2:

A^{2}_2+A^{2}_1+2A^{1}_1

Containing regular semisimple subalgebra number 3:

A^{2}_2+2A^{2}_1

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

3V_{4\psi}+8V_{3\psi}+11V_{2\psi}+8V_{\psi}+9V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&8h_{7}+16h_{6}+14h_{5}+12h_{4}+10h_{3}+8h_{2}+4h_{1}\\ e&=&g_{43}+g_{37}+2g_{34}+g_{29}+2g_{26}+g_{19}\\ f&=&g_{-19}+g_{-26}+g_{-29}+g_{-34}+g_{-37}+g_{-43}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&8h_{7}+16h_{6}+14h_{5}+12h_{4}+10h_{3}+8h_{2}+4h_{1}\\ [h, e]&=&2g_{43}+2g_{37}+4g_{34}+2g_{29}+4g_{26}+2g_{19}\\ [h, f]&=&-2g_{-19}-2g_{-26}-2g_{-29}-2g_{-34}-2g_{-37}-2g_{-43}\end{array}

Centralizer type:

2A^{4}_1+A^{3}_1

Killing form square of Cartan element dual to ambient long root: 32
Basis of the centralizer (dimension: 9):

h_{7}-2h_{1}

g_{1}+g_{-7}

g_{3}-g_{-3}

g_{4}-g_{-4}

g_{5}-g_{-5}

g_{7}+g_{-1}

g_{10}-g_{-10}

g_{11}-g_{-11}

g_{16}-g_{-16}

Basis of centralizer intersected with cartan (dimension: 1):

1/2h_{7}-h_{1}

Cartan of centralizer (dimension: 3):

1/2g_{1}+1/2h_{7}-h_{1}+1/2g_{-7}

g_{5}-g_{-5}

g_{3}-g_{-3}

Cartan-generating semisimple element:

g_{3}+g_{1}+h_{7}-2h_{1}-g_{-3}+g_{-7}

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 2 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ \end{pmatrix}

Characteristic polynomial ad H:

x^9-2x^7-7x^5-4x^3

Factorization of characteristic polynomial of ad H: (x )(x )(x )(x -2)(x +2)(x^2+1)(x^2+1)
Eigenvalues of ad H:

0

2

-2

\sqrt{-1}

-\sqrt{-1}

9 eigenvectors of ad H:

(1,1,0,0,0,0,0,0,0)

(0,0,1,0,0,0,0,0,0)

(0,0,0,0,1,0,0,0,0)

(−12,−14,0,0,0,1,0,0,0)

(0,1,0,0,0,0,0,0,0)

(0,0,0,−1,0,0,1,0,0)

(0,0,0,0,0,0,0,−1,1)

(0,0,0,−−1,0,0,1,0,0)

(0,0,0,0,0,0,0,−−1,1)

Centralizer type: 2A^{4}_1+A^{3}_1
Reductive components (3 total):
Scalar product computed:

\begin{pmatrix}1/24\\ \end{pmatrix}

Simple basis of Cartan of centralizer (1 total):

g_{1}+h_{7}-2h_{1}+g_{-7}

matching e:

g_{7}-1/4g_{1}-1/2h_{7}+h_{1}+g_{-1}-1/4g_{-7}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 2 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

Linear space basis of intersection of centralizer and ambient Cartan:

g_{1}+h_{7}-2h_{1}+g_{-7}

matching e:

g_{7}-1/4g_{1}-1/2h_{7}+h_{1}+g_{-1}-1/4g_{-7}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 2 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form:

\begin{pmatrix}96\\ \end{pmatrix}

Scalar product computed:

\begin{pmatrix}1/32\\ \end{pmatrix}

Simple basis of Cartan of centralizer (1 total):

\sqrt{-1}g_{5}-\sqrt{-1}g_{3}+\sqrt{-1}g_{-3}-\sqrt{-1}g_{-5}

matching e:

g_{16}+\sqrt{-1}g_{11}+\sqrt{-1}g_{10}-g_{4}+g_{-4}-\sqrt{-1}g_{-10}-\sqrt{-1}g_{-11}-g_{-16}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{-1} & \sqrt{-1} & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -\sqrt{-1} & 0 & 0 & 0 & 0 & \sqrt{-1}\\ 0 & 0 & 0 & -\sqrt{-1} & 0 & 0 & 0 & 0 & \sqrt{-1}\\ 0 & 0 & 0 & 0 & 0 & 0 & -\sqrt{-1} & -\sqrt{-1} & 0\\ \end{pmatrix}

Linear space basis of intersection of centralizer and ambient Cartan:

\sqrt{-1}g_{5}-\sqrt{-1}g_{3}+\sqrt{-1}g_{-3}-\sqrt{-1}g_{-5}

matching e:

g_{16}+\sqrt{-1}g_{11}+\sqrt{-1}g_{10}-g_{4}+g_{-4}-\sqrt{-1}g_{-10}-\sqrt{-1}g_{-11}-g_{-16}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{-1} & \sqrt{-1} & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -\sqrt{-1} & 0 & 0 & 0 & 0 & \sqrt{-1}\\ 0 & 0 & 0 & -\sqrt{-1} & 0 & 0 & 0 & 0 & \sqrt{-1}\\ 0 & 0 & 0 & 0 & 0 & 0 & -\sqrt{-1} & -\sqrt{-1} & 0\\ \end{pmatrix}

Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form:

\begin{pmatrix}128\\ \end{pmatrix}

Scalar product computed:

\begin{pmatrix}1/32\\ \end{pmatrix}

Simple basis of Cartan of centralizer (1 total):

-\sqrt{-1}g_{5}-\sqrt{-1}g_{3}+\sqrt{-1}g_{-3}+\sqrt{-1}g_{-5}

matching e:

g_{16}+\sqrt{-1}g_{11}-\sqrt{-1}g_{10}+g_{4}-g_{-4}+\sqrt{-1}g_{-10}-\sqrt{-1}g_{-11}-g_{-16}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{-1} & -\sqrt{-1} & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -\sqrt{-1} & 0 & 0 & 0 & 0 & -\sqrt{-1}\\ 0 & 0 & 0 & \sqrt{-1} & 0 & 0 & 0 & 0 & \sqrt{-1}\\ 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{-1} & -\sqrt{-1} & 0\\ \end{pmatrix}

Linear space basis of intersection of centralizer and ambient Cartan:

-\sqrt{-1}g_{5}-\sqrt{-1}g_{3}+\sqrt{-1}g_{-3}+\sqrt{-1}g_{-5}

matching e:

g_{16}+\sqrt{-1}g_{11}-\sqrt{-1}g_{10}+g_{4}-g_{-4}+\sqrt{-1}g_{-10}-\sqrt{-1}g_{-11}-g_{-16}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{-1} & -\sqrt{-1} & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -\sqrt{-1} & 0 & 0 & 0 & 0 & -\sqrt{-1}\\ 0 & 0 & 0 & \sqrt{-1} & 0 & 0 & 0 & 0 & \sqrt{-1}\\ 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{-1} & -\sqrt{-1} & 0\\ \end{pmatrix}

Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form:

\begin{pmatrix}128\\ \end{pmatrix}

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&8h_{7}+16h_{6}+14h_{5}+12h_{4}+10h_{3}+8h_{2}+4h_{1}\\ e&=&x_{3} g_{43}+x_{4} g_{37}+x_{1} g_{34}+x_{5} g_{29}+x_{2} g_{26}+x_{6} g_{19}\\ f&=&x_{12} g_{-19}+x_{8} g_{-26}+x_{11} g_{-29}+x_{7} g_{-34}+x_{10} g_{-37}+x_{9} g_{-43}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(2x_{1} x_{7} +x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -8\right)h_{7}+\left(2x_{1} x_{7} +2x_{2} x_{8} +2x_{3} x_{9} +2x_{4} x_{10} +2x_{5} x_{11} +2x_{6} x_{12} -16\right)h_{6}+\left(2x_{1} x_{7} +2x_{2} x_{8} +2x_{3} x_{9} +2x_{4} x_{10} +2x_{5} x_{11} -14\right)h_{5}+\left(2x_{1} x_{7} +2x_{2} x_{8} +2x_{3} x_{9} +2x_{4} x_{10} -12\right)h_{4}+\left(2x_{1} x_{7} +2x_{2} x_{8} +2x_{3} x_{9} -10\right)h_{3}+\left(2x_{1} x_{7} +2x_{2} x_{8} -8\right)h_{2}+\left(2x_{1} x_{7} -4\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{7} -4&=&0\\2x_{1} x_{7} +2x_{2} x_{8} -8&=&0\\2x_{1} x_{7} +2x_{2} x_{8} +2x_{3} x_{9} -10&=&0\\2x_{1} x_{7} +2x_{2} x_{8} +2x_{3} x_{9} +2x_{4} x_{10} -12&=&0\\2x_{1} x_{7} +2x_{2} x_{8} +2x_{3} x_{9} +2x_{4} x_{10} +2x_{5} x_{11} -14&=&0\\2x_{1} x_{7} +2x_{2} x_{8} +2x_{3} x_{9} +2x_{4} x_{10} +2x_{5} x_{11} +2x_{6} x_{12} -16&=&0\\2x_{1} x_{7} +x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -8&=&0\\\end{array}

\begin{array}{rcl}h&=&8h_{7}+16h_{6}+14h_{5}+12h_{4}+10h_{3}+8h_{2}+4h_{1}\\e&=&x_{3} g_{43}+x_{4} g_{37}+x_{1} g_{34}+x_{5} g_{29}+x_{2} g_{26}+x_{6} g_{19}\\f&=&g_{-19}+g_{-26}+g_{-29}+g_{-34}+g_{-37}+g_{-43}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0\\ 2 & 2 & 0 & 0 & 0 & 0\\ 2 & 2 & 2 & 0 & 0 & 0\\ 2 & 2 & 2 & 2 & 0 & 0\\ 2 & 2 & 2 & 2 & 2 & 0\\ 2 & 2 & 2 & 2 & 2 & 2\\ 2 & 0 & 1 & 1 & 1 & 1\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}4\\ 8\\ 10\\ 12\\ 14\\ 16\\ 8\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&8h_{7}+16h_{6}+14h_{5}+12h_{4}+10h_{3}+8h_{2}+4h_{1}\\ e&=&x_{3} g_{43}+x_{4} g_{37}+x_{1} g_{34}+x_{5} g_{29}+x_{2} g_{26}+x_{6} g_{19}\\ f&=&x_{12} g_{-19}+x_{8} g_{-26}+x_{11} g_{-29}+x_{7} g_{-34}+x_{10} g_{-37}+x_{9} g_{-43}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{7} -4&=&0\\2x_{1} x_{7} +2x_{2} x_{8} -8&=&0\\2x_{1} x_{7} +2x_{2} x_{8} +2x_{3} x_{9} -10&=&0\\2x_{1} x_{7} +2x_{2} x_{8} +2x_{3} x_{9} +2x_{4} x_{10} -12&=&0\\2x_{1} x_{7} +2x_{2} x_{8} +2x_{3} x_{9} +2x_{4} x_{10} +2x_{5} x_{11} -14&=&0\\2x_{1} x_{7} +2x_{2} x_{8} +2x_{3} x_{9} +2x_{4} x_{10} +2x_{5} x_{11} +2x_{6} x_{12} -16&=&0\\2x_{1} x_{7} +x_{3} x_{9} +x_{4} x_{10} +x_{5} x_{11} +x_{6} x_{12} -8&=&0\\\end{array}

A^{11}_1

h-characteristic: (2, 0, 1, 0, 0, 0, 0)
Length of the weight dual to h: 22
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1:

B^{1}_2+A^{1}_1

Containing regular semisimple subalgebra number 2:

C^{1}_3

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

V_{6\psi}+V_{4\psi}+8V_{3\psi}+3V_{2\psi}+8V_{\psi}+36V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&5h_{7}+10h_{6}+10h_{5}+10h_{4}+10h_{3}+8h_{2}+6h_{1}\\ e&=&4g_{47}+g_{43}+3g_{1}\\ f&=&g_{-1}+g_{-43}+g_{-47}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&5h_{7}+10h_{6}+10h_{5}+10h_{4}+10h_{3}+8h_{2}+6h_{1}\\ [h, e]&=&8g_{47}+2g_{43}+6g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-43}-2g_{-47}\end{array}

Centralizer type:

C_4

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&5h_{7}+10h_{6}+10h_{5}+10h_{4}+10h_{3}+8h_{2}+6h_{1}\\ e&=&x_{1} g_{47}+x_{3} g_{43}+x_{2} g_{1}\\ f&=&x_{5} g_{-1}+x_{6} g_{-43}+x_{4} g_{-47}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(x_{1} x_{4} +x_{3} x_{6} -5\right)h_{7}+\left(2x_{1} x_{4} +2x_{3} x_{6} -10\right)h_{6}+\left(2x_{1} x_{4} +2x_{3} x_{6} -10\right)h_{5}+\left(2x_{1} x_{4} +2x_{3} x_{6} -10\right)h_{4}+\left(2x_{1} x_{4} +2x_{3} x_{6} -10\right)h_{3}+\left(2x_{1} x_{4} -8\right)h_{2}+\left(2x_{2} x_{5} -6\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{4} -8&=&0\\2x_{1} x_{4} +2x_{3} x_{6} -10&=&0\\2x_{1} x_{4} +2x_{3} x_{6} -10&=&0\\2x_{1} x_{4} +2x_{3} x_{6} -10&=&0\\2x_{1} x_{4} +2x_{3} x_{6} -10&=&0\\x_{1} x_{4} +x_{3} x_{6} -5&=&0\\2x_{2} x_{5} -6&=&0\\\end{array}

\begin{array}{rcl}h&=&5h_{7}+10h_{6}+10h_{5}+10h_{4}+10h_{3}+8h_{2}+6h_{1}\\e&=&x_{1} g_{47}+x_{3} g_{43}+x_{2} g_{1}\\f&=&g_{-1}+g_{-43}+g_{-47}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0\\ 2 & 0 & 2\\ 2 & 0 & 2\\ 2 & 0 & 2\\ 2 & 0 & 2\\ 1 & 0 & 1\\ 0 & 2 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}8\\ 10\\ 10\\ 10\\ 10\\ 5\\ 6\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&5h_{7}+10h_{6}+10h_{5}+10h_{4}+10h_{3}+8h_{2}+6h_{1}\\ e&=&x_{1} g_{47}+x_{3} g_{43}+x_{2} g_{1}\\ f&=&x_{5} g_{-1}+x_{6} g_{-43}+x_{4} g_{-47}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{4} -8&=&0\\2x_{1} x_{4} +2x_{3} x_{6} -10&=&0\\2x_{1} x_{4} +2x_{3} x_{6} -10&=&0\\2x_{1} x_{4} +2x_{3} x_{6} -10&=&0\\2x_{1} x_{4} +2x_{3} x_{6} -10&=&0\\x_{1} x_{4} +x_{3} x_{6} -5&=&0\\2x_{2} x_{5} -6&=&0\\\end{array}

A^{11}_1

h-characteristic: (0, 1, 0, 0, 1, 0, 0)
Length of the weight dual to h: 22
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1:

A^{2}_2+3A^{1}_1

Containing regular semisimple subalgebra number 2:

A^{2}_2+A^{2}_1+A^{1}_1

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

3V_{4\psi}+6V_{3\psi}+11V_{2\psi}+12V_{\psi}+9V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&7h_{7}+14h_{6}+14h_{5}+12h_{4}+10h_{3}+8h_{2}+4h_{1}\\ e&=&g_{43}+2g_{38}+g_{37}+g_{29}+2g_{21}\\ f&=&g_{-21}+g_{-29}+g_{-37}+g_{-38}+g_{-43}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&7h_{7}+14h_{6}+14h_{5}+12h_{4}+10h_{3}+8h_{2}+4h_{1}\\ [h, e]&=&2g_{43}+4g_{38}+2g_{37}+2g_{29}+4g_{21}\\ [h, f]&=&-2g_{-21}-2g_{-29}-2g_{-37}-2g_{-38}-2g_{-43}\end{array}

Centralizer type:

A^{8}_1+A^{3}_1+A_1

Killing form square of Cartan element dual to ambient long root: 32
Basis of the centralizer (dimension: 9):

g_{-7}

h_{6}-h_{1}

h_{7}

g_{1}+g_{-19}

g_{3}-g_{-3}

g_{4}-g_{-4}

g_{7}

g_{10}-g_{-10}

g_{19}+g_{-1}

Basis of centralizer intersected with cartan (dimension: 2):

h_{6}-h_{1}

-h_{7}

Cartan of centralizer (dimension: 3):

g_{19}-g_{7}+g_{1}+h_{6}-h_{1}+g_{-1}-g_{-7}+g_{-19}

-2g_{19}-2g_{1}-h_{7}-2h_{6}+2h_{1}-2g_{-1}-2g_{-19}

g_{10}+g_{4}+g_{3}-g_{-3}-g_{-4}-g_{-10}

Cartan-generating semisimple element:

g_{19}+g_{10}+g_{7}+g_{4}+g_{3}+g_{1}+h_{7}+h_{6}-h_{1}+g_{-1}-g_{-3}-g_{-4}+g_{-7}-g_{-10}+g_{-19}

adjoint action:

\begin{pmatrix}-1 & -1 & 2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & -2\\ 1 & 0 & 0 & 1 & 0 & 0 & -1 & 0 & -1\\ 0 & 1 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & -1 & 0\\ 0 & 1 & -2 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & 1 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}

Characteristic polynomial ad H:

x^9-7x^7-5x^5+75x^3

Factorization of characteristic polynomial of ad H: (x )(x )(x )(x^2-5)(x^2-5)(x^2+3)
Eigenvalues of ad H:

0

\sqrt{5}

-\sqrt{5}

\sqrt{-3}

-\sqrt{-3}

9 eigenvectors of ad H:

(1,0,12,0,0,0,1,0,0)

(0,0,0,0,1,1,0,1,0)

(0,1,12,1,0,0,0,0,1)

(125−32,0,−125+12,0,0,0,1,0,0)

(0,−5+1,−125+12,125−32,0,0,0,0,1)

(−125−32,0,125+12,0,0,0,1,0,0)

(0,5+1,125+12,−125−32,0,0,0,0,1)

(0,0,0,0,−12−3−12,12−3−12,0,1,0)

(0,0,0,0,12−3−12,−12−3−12,0,1,0)

Centralizer type: A^{8}_1+A^{3}_1+A^{1}_1
Reductive components (3 total):
Scalar product computed:

\begin{pmatrix}1/8\\ \end{pmatrix}

Simple basis of Cartan of centralizer (1 total):

2/5\sqrt{5}g_{7}+1/5\sqrt{5}h_{7}+2/5\sqrt{5}g_{-7}

matching e:

g_{7}+\left(-1/2\sqrt{5}+1/2\right)h_{7}+\left(1/2\sqrt{5}-3/2\right)g_{-7}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}-2/5\sqrt{5} & -2/5\sqrt{5} & 4/5\sqrt{5} & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 2/5\sqrt{5} & 0 & 0 & 0 & 0 & 0 & -2/5\sqrt{5} & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2/5\sqrt{5} & -4/5\sqrt{5} & 0 & 0 & 0 & 2/5\sqrt{5} & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

Linear space basis of intersection of centralizer and ambient Cartan:

2/5\sqrt{5}g_{7}+1/5\sqrt{5}h_{7}+2/5\sqrt{5}g_{-7}

matching e:

g_{7}+\left(-1/2\sqrt{5}+1/2\right)h_{7}+\left(1/2\sqrt{5}-3/2\right)g_{-7}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}-2/5\sqrt{5} & -2/5\sqrt{5} & 4/5\sqrt{5} & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 2/5\sqrt{5} & 0 & 0 & 0 & 0 & 0 & -2/5\sqrt{5} & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2/5\sqrt{5} & -4/5\sqrt{5} & 0 & 0 & 0 & 2/5\sqrt{5} & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form:

\begin{pmatrix}32\\ \end{pmatrix}

Scalar product computed:

\begin{pmatrix}1/24\\ \end{pmatrix}

Simple basis of Cartan of centralizer (1 total):

2/5\sqrt{5}g_{19}+2/5\sqrt{5}g_{1}+1/5\sqrt{5}h_{7}+2/5\sqrt{5}h_{6}-2/5\sqrt{5}h_{1}+2/5\sqrt{5}g_{-1}+2/5\sqrt{5}g_{-19}

matching e:

g_{19}+\left(1/2\sqrt{5}-3/2\right)g_{1}+\left(-1/2\sqrt{5}+1/2\right)h_{7}+\left(-\sqrt{5}+1\right)h_{6}+\left(\sqrt{5}-1\right)h_{1}+g_{-1}+\left(1/2\sqrt{5}-3/2\right)g_{-19}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 4/5\sqrt{5} & 0 & 0 & 0 & 0 & -4/5\sqrt{5}\\ 0 & 0 & 0 & 2/5\sqrt{5} & 0 & 0 & 0 & 0 & -2/5\sqrt{5}\\ 0 & 2/5\sqrt{5} & 0 & -2/5\sqrt{5} & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -2/5\sqrt{5} & 0 & 0 & 0 & 0 & 0 & 0 & 2/5\sqrt{5}\\ \end{pmatrix}

Linear space basis of intersection of centralizer and ambient Cartan:

2/5\sqrt{5}g_{19}+2/5\sqrt{5}g_{1}+1/5\sqrt{5}h_{7}+2/5\sqrt{5}h_{6}-2/5\sqrt{5}h_{1}+2/5\sqrt{5}g_{-1}+2/5\sqrt{5}g_{-19}

matching e:

g_{19}+\left(1/2\sqrt{5}-3/2\right)g_{1}+\left(-1/2\sqrt{5}+1/2\right)h_{7}+\left(-\sqrt{5}+1\right)h_{6}+\left(\sqrt{5}-1\right)h_{1}+g_{-1}+\left(1/2\sqrt{5}-3/2\right)g_{-19}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 4/5\sqrt{5} & 0 & 0 & 0 & 0 & -4/5\sqrt{5}\\ 0 & 0 & 0 & 2/5\sqrt{5} & 0 & 0 & 0 & 0 & -2/5\sqrt{5}\\ 0 & 2/5\sqrt{5} & 0 & -2/5\sqrt{5} & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -2/5\sqrt{5} & 0 & 0 & 0 & 0 & 0 & 0 & 2/5\sqrt{5}\\ \end{pmatrix}

Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form:

\begin{pmatrix}96\\ \end{pmatrix}

Scalar product computed:

\begin{pmatrix}1/64\\ \end{pmatrix}

Simple basis of Cartan of centralizer (1 total):

-2/3\sqrt{-3}g_{10}-2/3\sqrt{-3}g_{4}-2/3\sqrt{-3}g_{3}+2/3\sqrt{-3}g_{-3}+2/3\sqrt{-3}g_{-4}+2/3\sqrt{-3}g_{-10}

matching e:

g_{10}+\left(1/2\sqrt{-3}-1/2\right)g_{4}+\left(-1/2\sqrt{-3}-1/2\right)g_{3}+\left(1/2\sqrt{-3}+1/2\right)g_{-3}+\left(-1/2\sqrt{-3}+1/2\right)g_{-4}-g_{-10}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2/3\sqrt{-3} & 0 & -2/3\sqrt{-3} & 0\\ 0 & 0 & 0 & 0 & -2/3\sqrt{-3} & 0 & 0 & 2/3\sqrt{-3} & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 2/3\sqrt{-3} & -2/3\sqrt{-3} & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

Linear space basis of intersection of centralizer and ambient Cartan:

-2/3\sqrt{-3}g_{10}-2/3\sqrt{-3}g_{4}-2/3\sqrt{-3}g_{3}+2/3\sqrt{-3}g_{-3}+2/3\sqrt{-3}g_{-4}+2/3\sqrt{-3}g_{-10}

matching e:

g_{10}+\left(1/2\sqrt{-3}-1/2\right)g_{4}+\left(-1/2\sqrt{-3}-1/2\right)g_{3}+\left(1/2\sqrt{-3}+1/2\right)g_{-3}+\left(-1/2\sqrt{-3}+1/2\right)g_{-4}-g_{-10}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2/3\sqrt{-3} & 0 & -2/3\sqrt{-3} & 0\\ 0 & 0 & 0 & 0 & -2/3\sqrt{-3} & 0 & 0 & 2/3\sqrt{-3} & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 2/3\sqrt{-3} & -2/3\sqrt{-3} & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form:

\begin{pmatrix}256\\ \end{pmatrix}

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&7h_{7}+14h_{6}+14h_{5}+12h_{4}+10h_{3}+8h_{2}+4h_{1}\\ e&=&x_{3} g_{43}+x_{1} g_{38}+x_{4} g_{37}+x_{5} g_{29}+x_{2} g_{21}\\ f&=&x_{7} g_{-21}+x_{10} g_{-29}+x_{9} g_{-37}+x_{6} g_{-38}+x_{8} g_{-43}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(2x_{1} x_{6} +x_{3} x_{8} +x_{4} x_{9} +x_{5} x_{10} -7\right)h_{7}+\left(4x_{1} x_{6} +2x_{3} x_{8} +2x_{4} x_{9} +2x_{5} x_{10} -14\right)h_{6}+\left(2x_{1} x_{6} +2x_{2} x_{7} +2x_{3} x_{8} +2x_{4} x_{9} +2x_{5} x_{10} -14\right)h_{5}+\left(2x_{1} x_{6} +2x_{2} x_{7} +2x_{3} x_{8} +2x_{4} x_{9} -12\right)h_{4}+\left(2x_{1} x_{6} +2x_{2} x_{7} +2x_{3} x_{8} -10\right)h_{3}+\left(2x_{1} x_{6} +2x_{2} x_{7} -8\right)h_{2}+\left(2x_{1} x_{6} -4\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{6} -4&=&0\\2x_{1} x_{6} +2x_{2} x_{7} -8&=&0\\2x_{1} x_{6} +2x_{2} x_{7} +2x_{3} x_{8} -10&=&0\\2x_{1} x_{6} +2x_{2} x_{7} +2x_{3} x_{8} +2x_{4} x_{9} -12&=&0\\2x_{1} x_{6} +2x_{2} x_{7} +2x_{3} x_{8} +2x_{4} x_{9} +2x_{5} x_{10} -14&=&0\\4x_{1} x_{6} +2x_{3} x_{8} +2x_{4} x_{9} +2x_{5} x_{10} -14&=&0\\2x_{1} x_{6} +x_{3} x_{8} +x_{4} x_{9} +x_{5} x_{10} -7&=&0\\\end{array}

\begin{array}{rcl}h&=&7h_{7}+14h_{6}+14h_{5}+12h_{4}+10h_{3}+8h_{2}+4h_{1}\\e&=&x_{3} g_{43}+x_{1} g_{38}+x_{4} g_{37}+x_{5} g_{29}+x_{2} g_{21}\\f&=&g_{-21}+g_{-29}+g_{-37}+g_{-38}+g_{-43}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0 & 0 & 0\\ 2 & 2 & 0 & 0 & 0\\ 2 & 2 & 2 & 0 & 0\\ 2 & 2 & 2 & 2 & 0\\ 2 & 2 & 2 & 2 & 2\\ 4 & 0 & 2 & 2 & 2\\ 2 & 0 & 1 & 1 & 1\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}4\\ 8\\ 10\\ 12\\ 14\\ 14\\ 7\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&7h_{7}+14h_{6}+14h_{5}+12h_{4}+10h_{3}+8h_{2}+4h_{1}\\ e&=&x_{3} g_{43}+x_{1} g_{38}+x_{4} g_{37}+x_{5} g_{29}+x_{2} g_{21}\\ f&=&x_{7} g_{-21}+x_{10} g_{-29}+x_{9} g_{-37}+x_{6} g_{-38}+x_{8} g_{-43}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{6} -4&=&0\\2x_{1} x_{6} +2x_{2} x_{7} -8&=&0\\2x_{1} x_{6} +2x_{2} x_{7} +2x_{3} x_{8} -10&=&0\\2x_{1} x_{6} +2x_{2} x_{7} +2x_{3} x_{8} +2x_{4} x_{9} -12&=&0\\2x_{1} x_{6} +2x_{2} x_{7} +2x_{3} x_{8} +2x_{4} x_{9} +2x_{5} x_{10} -14&=&0\\4x_{1} x_{6} +2x_{3} x_{8} +2x_{4} x_{9} +2x_{5} x_{10} -14&=&0\\2x_{1} x_{6} +x_{3} x_{8} +x_{4} x_{9} +x_{5} x_{10} -7&=&0\\\end{array}

A^{10}_1

h-characteristic: (2, 1, 0, 0, 0, 0, 0)
Length of the weight dual to h: 20
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1:

B^{1}_2

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

V_{6\psi}+10V_{3\psi}+V_{2\psi}+55V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&4h_{7}+8h_{6}+8h_{5}+8h_{4}+8h_{3}+8h_{2}+6h_{1}\\ e&=&4g_{47}+3g_{1}\\ f&=&g_{-1}+g_{-47}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&4h_{7}+8h_{6}+8h_{5}+8h_{4}+8h_{3}+8h_{2}+6h_{1}\\ [h, e]&=&8g_{47}+6g_{1}\\ [h, f]&=&-2g_{-1}-2g_{-47}\end{array}

Centralizer type:

C_5

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&4h_{7}+8h_{6}+8h_{5}+8h_{4}+8h_{3}+8h_{2}+6h_{1}\\ e&=&x_{1} g_{47}+x_{2} g_{1}\\ f&=&x_{4} g_{-1}+x_{3} g_{-47}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(x_{1} x_{3} -4\right)h_{7}+\left(2x_{1} x_{3} -8\right)h_{6}+\left(2x_{1} x_{3} -8\right)h_{5}+\left(2x_{1} x_{3} -8\right)h_{4}+\left(2x_{1} x_{3} -8\right)h_{3}+\left(2x_{1} x_{3} -8\right)h_{2}+\left(2x_{2} x_{4} -6\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{3} -8&=&0\\2x_{1} x_{3} -8&=&0\\2x_{1} x_{3} -8&=&0\\2x_{1} x_{3} -8&=&0\\2x_{1} x_{3} -8&=&0\\x_{1} x_{3} -4&=&0\\2x_{2} x_{4} -6&=&0\\\end{array}

\begin{array}{rcl}h&=&4h_{7}+8h_{6}+8h_{5}+8h_{4}+8h_{3}+8h_{2}+6h_{1}\\e&=&x_{1} g_{47}+x_{2} g_{1}\\f&=&g_{-1}+g_{-47}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0\\ 2 & 0\\ 2 & 0\\ 2 & 0\\ 2 & 0\\ 1 & 0\\ 0 & 2\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}8\\ 8\\ 8\\ 8\\ 8\\ 4\\ 6\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&4h_{7}+8h_{6}+8h_{5}+8h_{4}+8h_{3}+8h_{2}+6h_{1}\\ e&=&x_{1} g_{47}+x_{2} g_{1}\\ f&=&x_{4} g_{-1}+x_{3} g_{-47}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{3} -8&=&0\\2x_{1} x_{3} -8&=&0\\2x_{1} x_{3} -8&=&0\\2x_{1} x_{3} -8&=&0\\2x_{1} x_{3} -8&=&0\\x_{1} x_{3} -4&=&0\\2x_{2} x_{4} -6&=&0\\\end{array}

A^{10}_1

h-characteristic: (0, 1, 0, 1, 0, 0, 0)
Length of the weight dual to h: 20
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1:

A^{2}_2+2A^{1}_1

Containing regular semisimple subalgebra number 2:

A^{2}_2+A^{2}_1

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

3V_{4\psi}+4V_{3\psi}+12V_{2\psi}+12V_{\psi}+14V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&6h_{7}+12h_{6}+12h_{5}+12h_{4}+10h_{3}+8h_{2}+4h_{1}\\ e&=&g_{43}+2g_{41}+g_{37}+2g_{15}\\ f&=&g_{-15}+g_{-37}+g_{-41}+g_{-43}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&6h_{7}+12h_{6}+12h_{5}+12h_{4}+10h_{3}+8h_{2}+4h_{1}\\ [h, e]&=&2g_{43}+4g_{41}+2g_{37}+4g_{15}\\ [h, f]&=&-2g_{-15}-2g_{-37}-2g_{-41}-2g_{-43}\end{array}

Centralizer type:

B_2+A^{3}_1

Killing form square of Cartan element dual to ambient long root: 32
Basis of the centralizer (dimension: 14):

g_{-19}

g_{-13}

g_{-7}

g_{-6}

h_{5}-h_{1}

h_{6}

h_{7}

g_{1}+g_{-29}

g_{3}-g_{-3}

g_{6}

g_{7}

g_{13}

g_{19}

g_{29}+g_{-1}

Basis of centralizer intersected with cartan (dimension: 3):

h_{5}-h_{1}

-h_{7}

-h_{6}

Cartan of centralizer (dimension: 4):

g_{3}-g_{-3}

-h_{7}

-h_{6}

h_{5}-h_{1}

Cartan-generating semisimple element:

-h_{7}-h_{6}+h_{5}-h_{1}

adjoint action:

\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 3/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1/2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -3/2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}

Characteristic polynomial ad H:

x^{14}-17/2x^{12}+393/16x^{10}-239/8x^8+241/16x^6-9/4x^4

Factorization of characteristic polynomial of ad H: 1/16(x )(x )(x )(x )(x -2)(x -1)(x -1)(x +1)(x +1)(x +2)(2x -3)(2x -1)(2x +1)(2x +3)
Eigenvalues of ad H:

0

2

1

-1

-2

3/2

1/2

-1/2

-3/2

14 eigenvectors of ad H:

(0,0,0,0,1,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,1,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,1,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,1,0,0,0,0,0)

(1,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,1,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,1)

(0,0,0,0,0,0,0,1,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,1,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,1,0)

(0,1,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,1,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,1,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,1,0,0)

Centralizer type: B^{1}_2+A^{3}_1
Reductive components (2 total):
Scalar product computed:

\begin{pmatrix}1/8 & -1/16\\ -1/16 & 1/16\\ \end{pmatrix}

Simple basis of Cartan of centralizer (2 total):

-h_{7}

matching e:

g_{-7}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

-2h_{6}

matching e:

g_{-6}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

Linear space basis of intersection of centralizer and ambient Cartan:

-h_{7}

matching e:

g_{-7}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

-2h_{6}

matching e:

g_{-6}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

Elements in Cartan dual to root system: (1, 1), (-1, -1), (1, 0), (-1, 0), (2, 1), (-2, -1), (0, 1), (0, -1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form:

\begin{pmatrix}32 & -32\\ -32 & 64\\ \end{pmatrix}

Scalar product computed:

\begin{pmatrix}1/24\\ \end{pmatrix}

Simple basis of Cartan of centralizer (1 total):

h_{7}+2h_{6}+2h_{5}-2h_{1}

matching e:

g_{29}+g_{-1}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}

Linear space basis of intersection of centralizer and ambient Cartan:

h_{7}+2h_{6}+2h_{5}-2h_{1}

matching e:

g_{29}+g_{-1}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}

Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form:

\begin{pmatrix}96\\ \end{pmatrix}

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&6h_{7}+12h_{6}+12h_{5}+12h_{4}+10h_{3}+8h_{2}+4h_{1}\\ e&=&x_{3} g_{43}+x_{1} g_{41}+x_{4} g_{37}+x_{2} g_{15}\\ f&=&x_{6} g_{-15}+x_{8} g_{-37}+x_{5} g_{-41}+x_{7} g_{-43}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(2x_{1} x_{5} +x_{3} x_{7} +x_{4} x_{8} -6\right)h_{7}+\left(4x_{1} x_{5} +2x_{3} x_{7} +2x_{4} x_{8} -12\right)h_{6}+\left(4x_{1} x_{5} +2x_{3} x_{7} +2x_{4} x_{8} -12\right)h_{5}+\left(2x_{1} x_{5} +2x_{2} x_{6} +2x_{3} x_{7} +2x_{4} x_{8} -12\right)h_{4}+\left(2x_{1} x_{5} +2x_{2} x_{6} +2x_{3} x_{7} -10\right)h_{3}+\left(2x_{1} x_{5} +2x_{2} x_{6} -8\right)h_{2}+\left(2x_{1} x_{5} -4\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{5} -4&=&0\\2x_{1} x_{5} +2x_{2} x_{6} -8&=&0\\2x_{1} x_{5} +2x_{2} x_{6} +2x_{3} x_{7} -10&=&0\\2x_{1} x_{5} +2x_{2} x_{6} +2x_{3} x_{7} +2x_{4} x_{8} -12&=&0\\4x_{1} x_{5} +2x_{3} x_{7} +2x_{4} x_{8} -12&=&0\\4x_{1} x_{5} +2x_{3} x_{7} +2x_{4} x_{8} -12&=&0\\2x_{1} x_{5} +x_{3} x_{7} +x_{4} x_{8} -6&=&0\\\end{array}

\begin{array}{rcl}h&=&6h_{7}+12h_{6}+12h_{5}+12h_{4}+10h_{3}+8h_{2}+4h_{1}\\e&=&x_{3} g_{43}+x_{1} g_{41}+x_{4} g_{37}+x_{2} g_{15}\\f&=&g_{-15}+g_{-37}+g_{-41}+g_{-43}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0 & 0\\ 2 & 2 & 0 & 0\\ 2 & 2 & 2 & 0\\ 2 & 2 & 2 & 2\\ 4 & 0 & 2 & 2\\ 4 & 0 & 2 & 2\\ 2 & 0 & 1 & 1\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}4\\ 8\\ 10\\ 12\\ 12\\ 12\\ 6\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&6h_{7}+12h_{6}+12h_{5}+12h_{4}+10h_{3}+8h_{2}+4h_{1}\\ e&=&x_{3} g_{43}+x_{1} g_{41}+x_{4} g_{37}+x_{2} g_{15}\\ f&=&x_{6} g_{-15}+x_{8} g_{-37}+x_{5} g_{-41}+x_{7} g_{-43}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{5} -4&=&0\\2x_{1} x_{5} +2x_{2} x_{6} -8&=&0\\2x_{1} x_{5} +2x_{2} x_{6} +2x_{3} x_{7} -10&=&0\\2x_{1} x_{5} +2x_{2} x_{6} +2x_{3} x_{7} +2x_{4} x_{8} -12&=&0\\4x_{1} x_{5} +2x_{3} x_{7} +2x_{4} x_{8} -12&=&0\\4x_{1} x_{5} +2x_{3} x_{7} +2x_{4} x_{8} -12&=&0\\2x_{1} x_{5} +x_{3} x_{7} +x_{4} x_{8} -6&=&0\\\end{array}

A^{9}_1

h-characteristic: (0, 1, 1, 0, 0, 0, 0)
Length of the weight dual to h: 18
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1:

A^{2}_2+A^{1}_1

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

3V_{4\psi}+2V_{3\psi}+14V_{2\psi}+8V_{\psi}+24V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&5h_{7}+10h_{6}+10h_{5}+10h_{4}+10h_{3}+8h_{2}+4h_{1}\\ e&=&2g_{44}+g_{43}+2g_{9}\\ f&=&g_{-9}+g_{-43}+g_{-44}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&5h_{7}+10h_{6}+10h_{5}+10h_{4}+10h_{3}+8h_{2}+4h_{1}\\ [h, e]&=&4g_{44}+2g_{43}+4g_{9}\\ [h, f]&=&-2g_{-9}-2g_{-43}-2g_{-44}\end{array}

Centralizer type:

C_3+A^{3}_1

Killing form square of Cartan element dual to ambient long root: 32
Basis of the centralizer (dimension: 24):

g_{-29}

g_{-24}

g_{-19}

g_{-18}

g_{-13}

g_{-12}

g_{-7}

g_{-6}

g_{-5}

h_{4}-h_{1}

h_{5}

h_{6}

h_{7}

g_{1}+g_{-37}

g_{5}

g_{6}

g_{7}

g_{12}

g_{13}

g_{18}

g_{19}

g_{24}

g_{29}

g_{37}+g_{-1}

Basis of centralizer intersected with cartan (dimension: 4):

h_{4}-h_{1}

-h_{7}

-h_{6}

-h_{5}

Cartan of centralizer (dimension: 4):

-h_{7}

-h_{5}

h_{4}-h_{1}

-h_{6}

Cartan-generating semisimple element:

-5h_{7}+3h_{6}+4h_{5}+h_{4}-h_{1}

adjoint action:

\begin{pmatrix}-3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 7 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 13 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -6 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 6 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -13 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -7 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -5 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}

Characteristic polynomial ad H:

x^{24}-359x^{22}+45901x^{20}-2812259x^{18}+89775691x^{16}-1487336069x^{14}+12035802511x^{12}-45211399649x^{10}+77003894296x^8-59599451664x^6+17171481600x^4

Factorization of characteristic polynomial of ad H: (x )(x )(x )(x )(x -13)(x -8)(x -7)(x -6)(x -5)(x -3)(x -2)(x -1)(x -1)(x -1)(x +1)(x +1)(x +1)(x +2)(x +3)(x +5)(x +6)(x +7)(x +8)(x +13)
Eigenvalues of ad H:

0

13

8

7

6

5

3

2

1

-1

-2

-3

-5

-6

-7

-8

-13

24 eigenvectors of ad H:

(0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0)

(0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0)

(0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0)

(0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1)

(0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0)

(0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0)

(0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0)

(0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0)

Centralizer type: C^{1}_3+A^{3}_1
Reductive components (2 total):
Scalar product computed:

\begin{pmatrix}1/16 & -1/32 & 0\\ -1/32 & 1/16 & -1/16\\ 0 & -1/16 & 1/8\\ \end{pmatrix}

Simple basis of Cartan of centralizer (3 total):

-2h_{7}-2h_{6}-2h_{5}

matching e:

g_{-18}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

2h_{7}+4h_{6}+2h_{5}

matching e:

g_{24}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

-h_{7}-2h_{6}

matching e:

g_{-19}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

Linear space basis of intersection of centralizer and ambient Cartan:

-2h_{7}-2h_{6}-2h_{5}

matching e:

g_{-18}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

2h_{7}+4h_{6}+2h_{5}

matching e:

g_{24}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

-h_{7}-2h_{6}

matching e:

g_{-19}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

Elements in Cartan dual to root system: (1, 1, 1), (-1, -1, -1), (1, 2, 2), (-1, -2, -2), (1, 1, 2), (-1, -1, -2), (1, 1, 0), (-1, -1, 0), (1, 0, 0), (-1, 0, 0), (0, 1, 1), (0, -1, -1), (0, 1, 2), (0, -1, -2), (0, 0, 1), (0, 0, -1), (0, 1, 0), (0, -1, 0)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form:

\begin{pmatrix}64 & -32 & 0\\ -32 & 64 & -32\\ 0 & -32 & 32\\ \end{pmatrix}

Scalar product computed:

\begin{pmatrix}1/24\\ \end{pmatrix}

Simple basis of Cartan of centralizer (1 total):

h_{7}+2h_{6}+2h_{5}+2h_{4}-2h_{1}

matching e:

g_{37}+g_{-1}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}

Linear space basis of intersection of centralizer and ambient Cartan:

h_{7}+2h_{6}+2h_{5}+2h_{4}-2h_{1}

matching e:

g_{37}+g_{-1}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}

Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form:

\begin{pmatrix}96\\ \end{pmatrix}

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&5h_{7}+10h_{6}+10h_{5}+10h_{4}+10h_{3}+8h_{2}+4h_{1}\\ e&=&x_{1} g_{44}+x_{3} g_{43}+x_{2} g_{9}\\ f&=&x_{5} g_{-9}+x_{6} g_{-43}+x_{4} g_{-44}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(2x_{1} x_{4} +x_{3} x_{6} -5\right)h_{7}+\left(4x_{1} x_{4} +2x_{3} x_{6} -10\right)h_{6}+\left(4x_{1} x_{4} +2x_{3} x_{6} -10\right)h_{5}+\left(4x_{1} x_{4} +2x_{3} x_{6} -10\right)h_{4}+\left(2x_{1} x_{4} +2x_{2} x_{5} +2x_{3} x_{6} -10\right)h_{3}+\left(2x_{1} x_{4} +2x_{2} x_{5} -8\right)h_{2}+\left(2x_{1} x_{4} -4\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{4} -4&=&0\\2x_{1} x_{4} +2x_{2} x_{5} -8&=&0\\2x_{1} x_{4} +2x_{2} x_{5} +2x_{3} x_{6} -10&=&0\\4x_{1} x_{4} +2x_{3} x_{6} -10&=&0\\4x_{1} x_{4} +2x_{3} x_{6} -10&=&0\\4x_{1} x_{4} +2x_{3} x_{6} -10&=&0\\2x_{1} x_{4} +x_{3} x_{6} -5&=&0\\\end{array}

\begin{array}{rcl}h&=&5h_{7}+10h_{6}+10h_{5}+10h_{4}+10h_{3}+8h_{2}+4h_{1}\\e&=&x_{1} g_{44}+x_{3} g_{43}+x_{2} g_{9}\\f&=&g_{-9}+g_{-43}+g_{-44}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0\\ 2 & 2 & 0\\ 2 & 2 & 2\\ 4 & 0 & 2\\ 4 & 0 & 2\\ 4 & 0 & 2\\ 2 & 0 & 1\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}4\\ 8\\ 10\\ 10\\ 10\\ 10\\ 5\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&5h_{7}+10h_{6}+10h_{5}+10h_{4}+10h_{3}+8h_{2}+4h_{1}\\ e&=&x_{1} g_{44}+x_{3} g_{43}+x_{2} g_{9}\\ f&=&x_{5} g_{-9}+x_{6} g_{-43}+x_{4} g_{-44}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{4} -4&=&0\\2x_{1} x_{4} +2x_{2} x_{5} -8&=&0\\2x_{1} x_{4} +2x_{2} x_{5} +2x_{3} x_{6} -10&=&0\\4x_{1} x_{4} +2x_{3} x_{6} -10&=&0\\4x_{1} x_{4} +2x_{3} x_{6} -10&=&0\\4x_{1} x_{4} +2x_{3} x_{6} -10&=&0\\2x_{1} x_{4} +x_{3} x_{6} -5&=&0\\\end{array}

A^{8}_1

h-characteristic: (0, 2, 0, 0, 0, 0, 0)
Length of the weight dual to h: 16
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1:

A^{2}_2

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

3V_{4\psi}+17V_{2\psi}+39V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&4h_{7}+8h_{6}+8h_{5}+8h_{4}+8h_{3}+8h_{2}+4h_{1}\\ e&=&2g_{46}+2g_{2}\\ f&=&g_{-2}+g_{-46}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&4h_{7}+8h_{6}+8h_{5}+8h_{4}+8h_{3}+8h_{2}+4h_{1}\\ [h, e]&=&4g_{46}+4g_{2}\\ [h, f]&=&-2g_{-2}-2g_{-46}\end{array}

Centralizer type:

C_4+A^{3}_1

Killing form square of Cartan element dual to ambient long root: 32
Basis of the centralizer (dimension: 39):

g_{-37}

g_{-33}

g_{-29}

g_{-28}

g_{-24}

g_{-23}

g_{-19}

g_{-18}

g_{-17}

g_{-13}

g_{-12}

g_{-11}

g_{-7}

g_{-6}

g_{-5}

g_{-4}

h_{3}-h_{1}

h_{4}

h_{5}

h_{6}

h_{7}

g_{1}+g_{-43}

g_{4}

g_{5}

g_{6}

g_{7}

g_{11}

g_{12}

g_{13}

g_{17}

g_{18}

g_{19}

g_{23}

g_{24}

g_{28}

g_{29}

g_{33}

g_{37}

g_{43}+g_{-1}

Basis of centralizer intersected with cartan (dimension: 5):

h_{3}-h_{1}

-h_{7}

-h_{6}

-h_{5}

-h_{4}

Cartan of centralizer (dimension: 5):

-h_{6}

-h_{4}

-h_{5}

h_{3}-h_{1}

-h_{7}

Cartan-generating semisimple element:

-5h_{7}+3h_{6}+4h_{5}-2h_{4}+h_{3}-h_{1}

adjoint action:

\begin{pmatrix}3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -3/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -6 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -5/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 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& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -6 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -7/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 9/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -9/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 7/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 6 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -13 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 19/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -7 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -7/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -8 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 5/2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 6 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3/2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -3 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}

Characteristic polynomial ad H:

x^{39}-1077/2x^{37}+1941663/16x^{35}-122548019/8x^{33}+311601801951/256x^{31}-33095589371577/512x^{29}+9714579397013265/4096x^{27}-250699920070363515/4096x^{25}+4584675099477776985/4096x^{23}-59385823787146327735/4096x^{21}+540731611713501148359/4096x^{19}-3410082746090997519009/4096x^{17}+14559847861563998422147/4096x^{15}-40781421619797619817397/4096x^{13}+17966868824328980326551/1024x^{11}-4700498649698462115519/256x^9+660180953606071928625/64x^7-2384060216418050625x^5

Factorization of characteristic polynomial of ad H: 1/4096(x )(x )(x )(x )(x )(x -13)(x -8)(x -7)(x -6)(x -6)(x -5)(x -3)(x -2)(x -1)(x -1)(x -1)(x +1)(x +1)(x +1)(x +2)(x +3)(x +5)(x +6)(x +6)(x +7)(x +8)(x +13)(2x -19)(2x -9)(2x -7)(2x -7)(2x -5)(2x -3)(2x +3)(2x +5)(2x +7)(2x +7)(2x +9)(2x +19)
Eigenvalues of ad H:

0

13

8

7

6

5

3

2

1

-1

-2

-3

-5

-6

-7

-8

-13

19/2

9/2

7/2

5/2

3/2

-3/2

-5/2

-7/2

-9/2

-19/2

39 eigenvectors of ad H:

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0)

(1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0)

(0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0)

(0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

Centralizer type: C^{1}_4+A^{3}_1
Reductive components (2 total):
Scalar product computed:

\begin{pmatrix}1/8 & 0 & -1/16 & 0\\ 0 & 1/16 & -1/32 & -1/32\\ -1/16 & -1/32 & 1/16 & 0\\ 0 & -1/32 & 0 & 1/16\\ \end{pmatrix}

Simple basis of Cartan of centralizer (4 total):

-h_{7}-2h_{6}

matching e:

g_{-19}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 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2h_{7}+4h_{6}+4h_{5}+2h_{4}

matching e:

g_{33}

verification:

[h,e]-2e=0

adjoint action:

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-2h_{5}-2h_{4}

matching e:

g_{-11}

verification:

[h,e]-2e=0

adjoint action:

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-2h_{7}-2h_{6}-2h_{5}

matching e:

g_{-18}

verification:

[h,e]-2e=0

adjoint action:

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Linear space basis of intersection of centralizer and ambient Cartan:

-h_{7}-2h_{6}

matching e:

g_{-19}

verification:

[h,e]-2e=0

adjoint action:

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2h_{7}+4h_{6}+4h_{5}+2h_{4}

matching e:

g_{33}

verification:

[h,e]-2e=0

adjoint action:

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-2h_{5}-2h_{4}

matching e:

g_{-11}

verification:

[h,e]-2e=0

adjoint action:

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-2h_{7}-2h_{6}-2h_{5}

matching e:

g_{-18}

verification:

[h,e]-2e=0

adjoint action:

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Elements in Cartan dual to root system: (1, 1, 1, 1), (-1, -1, -1, -1), (2, 1, 2, 1), (-2, -1, -2, -1), (2, 1, 1, 1), (-2, -1, -1, -1), (0, 1, 1, 1), (0, -1, -1, -1), (0, 1, 0, 1), (0, -1, 0, -1), (1, 0, 1, 0), (-1, 0, -1, 0), (2, 0, 1, 0), (-2, 0, -1, 0), (0, 0, 1, 0), (0, 0, -1, 0), (1, 0, 0, 0), (-1, 0, 0, 0), (1, 1, 1, 0), (-1, -1, -1, 0), (2, 2, 2, 1), (-2, -2, -2, -1), (2, 1, 2, 0), (-2, -1, -2, 0), (0, 0, 0, 1), (0, 0, 0, -1), (2, 1, 1, 0), (-2, -1, -1, 0), (0, 1, 1, 0), (0, -1, -1, 0), (0, 1, 0, 0), (0, -1, 0, 0)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form:

\begin{pmatrix}32 & 0 & -32 & 0\\ 0 & 64 & -32 & -32\\ -32 & -32 & 64 & 0\\ 0 & -32 & 0 & 64\\ \end{pmatrix}

Scalar product computed:

\begin{pmatrix}1/24\\ \end{pmatrix}

Simple basis of Cartan of centralizer (1 total):

h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}-2h_{1}

matching e:

g_{43}+g_{-1}

verification:

[h,e]-2e=0

adjoint action:

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Linear space basis of intersection of centralizer and ambient Cartan:

h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}-2h_{1}

matching e:

g_{43}+g_{-1}

verification:

[h,e]-2e=0

adjoint action:

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& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}

Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form:

\begin{pmatrix}96\\ \end{pmatrix}

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&4h_{7}+8h_{6}+8h_{5}+8h_{4}+8h_{3}+8h_{2}+4h_{1}\\ e&=&x_{1} g_{46}+x_{2} g_{2}\\ f&=&x_{4} g_{-2}+x_{3} g_{-46}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(2x_{1} x_{3} -4\right)h_{7}+\left(4x_{1} x_{3} -8\right)h_{6}+\left(4x_{1} x_{3} -8\right)h_{5}+\left(4x_{1} x_{3} -8\right)h_{4}+\left(4x_{1} x_{3} -8\right)h_{3}+\left(2x_{1} x_{3} +2x_{2} x_{4} -8\right)h_{2}+\left(2x_{1} x_{3} -4\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{3} -4&=&0\\2x_{1} x_{3} +2x_{2} x_{4} -8&=&0\\4x_{1} x_{3} -8&=&0\\4x_{1} x_{3} -8&=&0\\4x_{1} x_{3} -8&=&0\\4x_{1} x_{3} -8&=&0\\2x_{1} x_{3} -4&=&0\\\end{array}

\begin{array}{rcl}h&=&4h_{7}+8h_{6}+8h_{5}+8h_{4}+8h_{3}+8h_{2}+4h_{1}\\e&=&x_{1} g_{46}+x_{2} g_{2}\\f&=&g_{-2}+g_{-46}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0\\ 2 & 2\\ 4 & 0\\ 4 & 0\\ 4 & 0\\ 4 & 0\\ 2 & 0\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}4\\ 8\\ 8\\ 8\\ 8\\ 8\\ 4\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&4h_{7}+8h_{6}+8h_{5}+8h_{4}+8h_{3}+8h_{2}+4h_{1}\\ e&=&x_{1} g_{46}+x_{2} g_{2}\\ f&=&x_{4} g_{-2}+x_{3} g_{-46}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{3} -4&=&0\\2x_{1} x_{3} +2x_{2} x_{4} -8&=&0\\4x_{1} x_{3} -8&=&0\\4x_{1} x_{3} -8&=&0\\4x_{1} x_{3} -8&=&0\\4x_{1} x_{3} -8&=&0\\2x_{1} x_{3} -4&=&0\\\end{array}

A^{7}_1

h-characteristic: (0, 0, 0, 0, 0, 0, 2)
Length of the weight dual to h: 14
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 4
Containing regular semisimple subalgebra number 1:

3A^{2}_1+A^{1}_1

Containing regular semisimple subalgebra number 2:

7A^{1}_1

Containing regular semisimple subalgebra number 3:

A^{2}_1+5A^{1}_1

Containing regular semisimple subalgebra number 4:

2A^{2}_1+3A^{1}_1

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

28V_{2\psi}+21V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&7h_{7}+12h_{6}+10h_{5}+8h_{4}+6h_{3}+4h_{2}+2h_{1}\\ e&=&g_{48}+g_{40}+g_{24}+g_{7}\\ f&=&g_{-7}+g_{-24}+g_{-40}+g_{-48}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&7h_{7}+12h_{6}+10h_{5}+8h_{4}+6h_{3}+4h_{2}+2h_{1}\\ [h, e]&=&2g_{48}+2g_{40}+2g_{24}+2g_{7}\\ [h, f]&=&-2g_{-7}-2g_{-24}-2g_{-40}-2g_{-48}\end{array}

Centralizer type:

B^{4}_3

Killing form square of Cartan element dual to ambient long root: 32
Basis of the centralizer (dimension: 21):

h_{1}

h_{3}

h_{5}

g_{2}-g_{-14}

g_{4}-g_{-16}

g_{6}-g_{-12}

g_{8}-g_{-9}

g_{9}-g_{-8}

g_{10}-g_{-11}

g_{11}-g_{-10}

g_{12}-g_{-6}

g_{14}-g_{-2}

g_{15}-g_{-25}

g_{16}-g_{-4}

g_{17}-g_{-22}

g_{20}-g_{-21}

g_{21}-g_{-20}

g_{22}-g_{-17}

g_{25}-g_{-15}

g_{26}-g_{-30}

g_{30}-g_{-26}

Basis of centralizer intersected with cartan (dimension: 3):

-h_{5}

-h_{3}

-h_{1}

Cartan of centralizer (dimension: 3):

-h_{5}

-h_{3}

-h_{1}

Cartan-generating semisimple element:

-h_{5}-5h_{3}+3h_{1}

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -3 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 5/2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -5/2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -3/2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3/2\\ \end{pmatrix}

Characteristic polynomial ad H:

x^{21}-175/4x^{19}+11991/16x^{17}-423125/64x^{15}+2110199/64x^{13}-6108375/64x^{11}+10091329/64x^9-1121225/8x^7+235809/4x^5-8100x^3

Factorization of characteristic polynomial of ad H: 1/64(x )(x )(x )(x -4)(x -3)(x -2)(x -2)(x -1)(x -1)(x +1)(x +1)(x +2)(x +2)(x +3)(x +4)(2x -5)(2x -3)(2x -1)(2x +1)(2x +3)(2x +5)
Eigenvalues of ad H:

0

4

3

2

1

-1

-2

-3

-4

5/2

3/2

1/2

-1/2

-3/2

-5/2

21 eigenvectors of ad H:

(1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0)

(0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0)

(0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1)

(0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0)

Centralizer type: B^{4}_3
Reductive components (1 total):
Scalar product computed:

\begin{pmatrix}1/32 & -1/64 & -1/64\\ -1/64 & 1/64 & 0\\ -1/64 & 0 & 1/32\\ \end{pmatrix}

Simple basis of Cartan of centralizer (3 total):

2h_{5}+2h_{1}

matching e:

g_{25}-g_{-15}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}

-4h_{5}

matching e:

g_{6}-g_{-12}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

-2h_{3}-2h_{1}

matching e:

g_{2}-g_{-14}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}

Linear space basis of intersection of centralizer and ambient Cartan:

2h_{5}+2h_{1}

matching e:

g_{25}-g_{-15}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}

-4h_{5}

matching e:

g_{6}-g_{-12}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

-2h_{3}-2h_{1}

matching e:

g_{2}-g_{-14}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}

Elements in Cartan dual to root system: (2, 1, 1), (-2, -1, -1), (1, 1, 1), (-1, -1, -1), (1, 0, 1), (-1, 0, -1), (1, 1, 0), (-1, -1, 0), (1, 0, 0), (-1, 0, 0), (0, 0, 1), (0, 0, -1), (2, 1, 2), (-2, -1, -2), (2, 1, 0), (-2, -1, 0), (0, 1, 0), (0, -1, 0)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form:

\begin{pmatrix}128 & -128 & -64\\ -128 & 256 & 0\\ -64 & 0 & 128\\ \end{pmatrix}

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&7h_{7}+12h_{6}+10h_{5}+8h_{4}+6h_{3}+4h_{2}+2h_{1}\\ e&=&x_{1} g_{48}+x_{2} g_{40}+x_{3} g_{24}+x_{4} g_{7}\\ f&=&x_{8} g_{-7}+x_{7} g_{-24}+x_{6} g_{-40}+x_{5} g_{-48}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(2x_{1} x_{5} +2x_{2} x_{6} +2x_{3} x_{7} +x_{4} x_{8} -7\right)h_{7}+\left(4x_{1} x_{5} +4x_{2} x_{6} +4x_{3} x_{7} -12\right)h_{6}+\left(4x_{1} x_{5} +4x_{2} x_{6} +2x_{3} x_{7} -10\right)h_{5}+\left(4x_{1} x_{5} +4x_{2} x_{6} -8\right)h_{4}+\left(4x_{1} x_{5} +2x_{2} x_{6} -6\right)h_{3}+\left(4x_{1} x_{5} -4\right)h_{2}+\left(2x_{1} x_{5} -2\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{5} -2&=&0\\4x_{1} x_{5} -4&=&0\\4x_{1} x_{5} +2x_{2} x_{6} -6&=&0\\4x_{1} x_{5} +4x_{2} x_{6} -8&=&0\\4x_{1} x_{5} +4x_{2} x_{6} +2x_{3} x_{7} -10&=&0\\4x_{1} x_{5} +4x_{2} x_{6} +4x_{3} x_{7} -12&=&0\\2x_{1} x_{5} +2x_{2} x_{6} +2x_{3} x_{7} +x_{4} x_{8} -7&=&0\\\end{array}

\begin{array}{rcl}h&=&7h_{7}+12h_{6}+10h_{5}+8h_{4}+6h_{3}+4h_{2}+2h_{1}\\e&=&x_{1} g_{48}+x_{2} g_{40}+x_{3} g_{24}+x_{4} g_{7}\\f&=&g_{-7}+g_{-24}+g_{-40}+g_{-48}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0 & 0\\ 4 & 0 & 0 & 0\\ 4 & 2 & 0 & 0\\ 4 & 4 & 0 & 0\\ 4 & 4 & 2 & 0\\ 4 & 4 & 4 & 0\\ 2 & 2 & 2 & 1\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}2\\ 4\\ 6\\ 8\\ 10\\ 12\\ 7\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&7h_{7}+12h_{6}+10h_{5}+8h_{4}+6h_{3}+4h_{2}+2h_{1}\\ e&=&x_{1} g_{48}+x_{2} g_{40}+x_{3} g_{24}+x_{4} g_{7}\\ f&=&x_{8} g_{-7}+x_{7} g_{-24}+x_{6} g_{-40}+x_{5} g_{-48}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{5} -2&=&0\\4x_{1} x_{5} -4&=&0\\4x_{1} x_{5} +2x_{2} x_{6} -6&=&0\\4x_{1} x_{5} +4x_{2} x_{6} -8&=&0\\4x_{1} x_{5} +4x_{2} x_{6} +2x_{3} x_{7} -10&=&0\\4x_{1} x_{5} +4x_{2} x_{6} +4x_{3} x_{7} -12&=&0\\2x_{1} x_{5} +2x_{2} x_{6} +2x_{3} x_{7} +x_{4} x_{8} -7&=&0\\\end{array}

A^{6}_1

h-characteristic: (0, 0, 0, 0, 0, 1, 0)
Length of the weight dual to h: 12
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 4
Containing regular semisimple subalgebra number 1:

2A^{2}_1+2A^{1}_1

Containing regular semisimple subalgebra number 2:

3A^{2}_1

Containing regular semisimple subalgebra number 3:

6A^{1}_1

Containing regular semisimple subalgebra number 4:

A^{2}_1+4A^{1}_1

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

21V_{2\psi}+12V_{\psi}+18V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&6h_{7}+12h_{6}+10h_{5}+8h_{4}+6h_{3}+4h_{2}+2h_{1}\\ e&=&g_{48}+g_{40}+g_{29}+g_{19}\\ f&=&g_{-19}+g_{-29}+g_{-40}+g_{-48}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&6h_{7}+12h_{6}+10h_{5}+8h_{4}+6h_{3}+4h_{2}+2h_{1}\\ [h, e]&=&2g_{48}+2g_{40}+2g_{29}+2g_{19}\\ [h, f]&=&-2g_{-19}-2g_{-29}-2g_{-40}-2g_{-48}\end{array}

Centralizer type:

A^{4}_3+A_1

Killing form square of Cartan element dual to ambient long root: 32
Basis of the centralizer (dimension: 18):

g_{-7}

h_{1}

h_{3}

h_{7}

g_{2}-g_{-14}

g_{4}-g_{-10}

g_{5}-g_{-5}

g_{7}

g_{8}-g_{-9}

g_{9}-g_{-8}

g_{10}-g_{-4}

g_{11}-g_{-16}

g_{14}-g_{-2}

g_{15}-g_{-20}

g_{16}-g_{-11}

g_{20}-g_{-15}

g_{21}-g_{-25}

g_{25}-g_{-21}

Basis of centralizer intersected with cartan (dimension: 3):

-h_{1}

-h_{7}

-h_{3}

Cartan of centralizer (dimension: 4):

g_{5}-g_{-5}

-h_{7}

-h_{3}

-h_{1}

Cartan-generating semisimple element:

-5h_{7}+3h_{3}-h_{1}

adjoint action:

\begin{pmatrix}10 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -3/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -10 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -3/2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1/2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3/2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1/2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1/2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1/2\\ \end{pmatrix}

Characteristic polynomial ad H:

x^{18}-110x^{16}+8291/8x^{14}-59155/16x^{12}+1539233/256x^{10}-1126585/256x^8+41103/32x^6-2025/16x^4

Factorization of characteristic polynomial of ad H: 1/256(x )(x )(x )(x )(x -10)(x -2)(x -1)(x +1)(x +2)(x +10)(2x -3)(2x -3)(2x -1)(2x -1)(2x +1)(2x +1)(2x +3)(2x +3)
Eigenvalues of ad H:

0

10

2

1

-1

-2

-10

3/2

1/2

-1/2

-3/2

18 eigenvectors of ad H:

(0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0)

(1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0)

(0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1)

(0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0)

Centralizer type: A^{4}_3+A^{1}_1
Reductive components (2 total):
Scalar product computed:

\begin{pmatrix}1/8\\ \end{pmatrix}

Simple basis of Cartan of centralizer (1 total):

-h_{7}

matching e:

g_{-7}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

Linear space basis of intersection of centralizer and ambient Cartan:

-h_{7}

matching e:

g_{-7}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form:

\begin{pmatrix}32\\ \end{pmatrix}

Scalar product computed:

\begin{pmatrix}1/32 & 0 & -1/64\\ 0 & 1/32 & -1/64\\ -1/64 & -1/64 & 1/32\\ \end{pmatrix}

Simple basis of Cartan of centralizer (3 total):

\sqrt{-1}g_{5}-2h_{1}-\sqrt{-1}g_{-5}

matching e:

g_{21}+\sqrt{-1}g_{15}-\sqrt{-1}g_{-20}-g_{-25}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{-1} & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{-1} & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -\sqrt{-1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & \sqrt{-1} & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -\sqrt{-1} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & \sqrt{-1}\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -\sqrt{-1} & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -\sqrt{-1} & 0 & -1\\ \end{pmatrix}

-\sqrt{-1}g_{5}-2h_{1}+\sqrt{-1}g_{-5}

matching e:

g_{21}-\sqrt{-1}g_{15}+\sqrt{-1}g_{-20}-g_{-25}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -\sqrt{-1} & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -\sqrt{-1} & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & \sqrt{-1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & -\sqrt{-1} & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{-1} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & -\sqrt{-1}\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{-1} & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{-1} & 0 & -1\\ \end{pmatrix}

2h_{3}+2h_{1}

matching e:

g_{14}-g_{-2}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}

Linear space basis of intersection of centralizer and ambient Cartan:

\sqrt{-1}g_{5}-2h_{1}-\sqrt{-1}g_{-5}

matching e:

g_{21}+\sqrt{-1}g_{15}-\sqrt{-1}g_{-20}-g_{-25}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{-1} & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{-1} & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -\sqrt{-1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & \sqrt{-1} & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -\sqrt{-1} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & \sqrt{-1}\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -\sqrt{-1} & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -\sqrt{-1} & 0 & -1\\ \end{pmatrix}

-\sqrt{-1}g_{5}-2h_{1}+\sqrt{-1}g_{-5}

matching e:

g_{21}-\sqrt{-1}g_{15}+\sqrt{-1}g_{-20}-g_{-25}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -\sqrt{-1} & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -\sqrt{-1} & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & \sqrt{-1} & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & -\sqrt{-1} & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{-1} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & -\sqrt{-1}\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{-1} & 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{-1} & 0 & -1\\ \end{pmatrix}

2h_{3}+2h_{1}

matching e:

g_{14}-g_{-2}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}

Elements in Cartan dual to root system: (1, 1, 1), (-1, -1, -1), (0, 0, 1), (0, 0, -1), (1, 0, 1), (-1, 0, -1), (0, 1, 1), (0, -1, -1), (1, 0, 0), (-1, 0, 0), (0, 1, 0), (0, -1, 0)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form:

\begin{pmatrix}128 & 0 & -64\\ 0 & 128 & -64\\ -64 & -64 & 128\\ \end{pmatrix}

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&6h_{7}+12h_{6}+10h_{5}+8h_{4}+6h_{3}+4h_{2}+2h_{1}\\ e&=&x_{1} g_{48}+x_{2} g_{40}+x_{3} g_{29}+x_{4} g_{19}\\ f&=&x_{8} g_{-19}+x_{7} g_{-29}+x_{6} g_{-40}+x_{5} g_{-48}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(2x_{1} x_{5} +2x_{2} x_{6} +x_{3} x_{7} +x_{4} x_{8} -6\right)h_{7}+\left(4x_{1} x_{5} +4x_{2} x_{6} +2x_{3} x_{7} +2x_{4} x_{8} -12\right)h_{6}+\left(4x_{1} x_{5} +4x_{2} x_{6} +2x_{3} x_{7} -10\right)h_{5}+\left(4x_{1} x_{5} +4x_{2} x_{6} -8\right)h_{4}+\left(4x_{1} x_{5} +2x_{2} x_{6} -6\right)h_{3}+\left(4x_{1} x_{5} -4\right)h_{2}+\left(2x_{1} x_{5} -2\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{5} -2&=&0\\4x_{1} x_{5} -4&=&0\\4x_{1} x_{5} +2x_{2} x_{6} -6&=&0\\4x_{1} x_{5} +4x_{2} x_{6} -8&=&0\\4x_{1} x_{5} +4x_{2} x_{6} +2x_{3} x_{7} -10&=&0\\4x_{1} x_{5} +4x_{2} x_{6} +2x_{3} x_{7} +2x_{4} x_{8} -12&=&0\\2x_{1} x_{5} +2x_{2} x_{6} +x_{3} x_{7} +x_{4} x_{8} -6&=&0\\\end{array}

\begin{array}{rcl}h&=&6h_{7}+12h_{6}+10h_{5}+8h_{4}+6h_{3}+4h_{2}+2h_{1}\\e&=&x_{1} g_{48}+x_{2} g_{40}+x_{3} g_{29}+x_{4} g_{19}\\f&=&g_{-19}+g_{-29}+g_{-40}+g_{-48}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0 & 0\\ 4 & 0 & 0 & 0\\ 4 & 2 & 0 & 0\\ 4 & 4 & 0 & 0\\ 4 & 4 & 2 & 0\\ 4 & 4 & 2 & 2\\ 2 & 2 & 1 & 1\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}2\\ 4\\ 6\\ 8\\ 10\\ 12\\ 6\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&6h_{7}+12h_{6}+10h_{5}+8h_{4}+6h_{3}+4h_{2}+2h_{1}\\ e&=&x_{1} g_{48}+x_{2} g_{40}+x_{3} g_{29}+x_{4} g_{19}\\ f&=&x_{8} g_{-19}+x_{7} g_{-29}+x_{6} g_{-40}+x_{5} g_{-48}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{5} -2&=&0\\4x_{1} x_{5} -4&=&0\\4x_{1} x_{5} +2x_{2} x_{6} -6&=&0\\4x_{1} x_{5} +4x_{2} x_{6} -8&=&0\\4x_{1} x_{5} +4x_{2} x_{6} +2x_{3} x_{7} -10&=&0\\4x_{1} x_{5} +4x_{2} x_{6} +2x_{3} x_{7} +2x_{4} x_{8} -12&=&0\\2x_{1} x_{5} +2x_{2} x_{6} +x_{3} x_{7} +x_{4} x_{8} -6&=&0\\\end{array}

A^{5}_1

h-characteristic: (0, 0, 0, 0, 1, 0, 0)
Length of the weight dual to h: 10
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1:

2A^{2}_1+A^{1}_1

Containing regular semisimple subalgebra number 2:

5A^{1}_1

Containing regular semisimple subalgebra number 3:

A^{2}_1+3A^{1}_1

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

15V_{2\psi}+20V_{\psi}+20V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&5h_{7}+10h_{6}+10h_{5}+8h_{4}+6h_{3}+4h_{2}+2h_{1}\\ e&=&g_{48}+g_{40}+g_{29}\\ f&=&g_{-29}+g_{-40}+g_{-48}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&5h_{7}+10h_{6}+10h_{5}+8h_{4}+6h_{3}+4h_{2}+2h_{1}\\ [h, e]&=&2g_{48}+2g_{40}+2g_{29}\\ [h, f]&=&-2g_{-29}-2g_{-40}-2g_{-48}\end{array}

Centralizer type:

B^{4}_2+B_2

Killing form square of Cartan element dual to ambient long root: 32
Basis of the centralizer (dimension: 20):

g_{-19}

g_{-13}

g_{-7}

g_{-6}

h_{1}

h_{3}

h_{6}

h_{7}

g_{2}-g_{-14}

g_{4}-g_{-10}

g_{6}

g_{7}

g_{8}-g_{-9}

g_{9}-g_{-8}

g_{10}-g_{-4}

g_{13}

g_{14}-g_{-2}

g_{15}-g_{-20}

g_{19}

g_{20}-g_{-15}

Basis of centralizer intersected with cartan (dimension: 4):

-h_{7}

-h_{6}

-h_{3}

-h_{1}

Cartan of centralizer (dimension: 4):

-h_{6}

-h_{3}

-h_{1}

-h_{7}

Cartan-generating semisimple element:

-h_{7}-5h_{6}+3h_{3}-2h_{1}

adjoint action:

\begin{pmatrix}5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -3/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -5/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 5/2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3/2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1/2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -5 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}

Characteristic polynomial ad H:

x^{20}-243/4x^{18}+21459/16x^{16}-873329/64x^{14}+1082079/16x^{12}-5131779/32x^{10}+714949/4x^8-5522625/64x^6+50625/4x^4

Factorization of characteristic polynomial of ad H: 1/64(x )(x )(x )(x )(x -5)(x -4)(x -3)(x -1)(x -1)(x +1)(x +1)(x +3)(x +4)(x +5)(2x -5)(2x -3)(2x -1)(2x +1)(2x +3)(2x +5)
Eigenvalues of ad H:

0

5

4

3

1

-1

-3

-4

-5

5/2

3/2

1/2

-1/2

-3/2

-5/2

20 eigenvectors of ad H:

(0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0)

(1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0)

(0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1)

(0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0)

(0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0)

Centralizer type: B^{4}_2+B^{1}_2
Reductive components (2 total):
Scalar product computed:

\begin{pmatrix}1/8 & -1/16\\ -1/16 & 1/16\\ \end{pmatrix}

Simple basis of Cartan of centralizer (2 total):

h_{7}

matching e:

g_{7}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

-2h_{7}-2h_{6}

matching e:

g_{-13}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

Linear space basis of intersection of centralizer and ambient Cartan:

h_{7}

matching e:

g_{7}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

-2h_{7}-2h_{6}

matching e:

g_{-13}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

Elements in Cartan dual to root system: (1, 1), (-1, -1), (2, 1), (-2, -1), (1, 0), (-1, 0), (0, 1), (0, -1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form:

\begin{pmatrix}32 & -32\\ -32 & 64\\ \end{pmatrix}

Scalar product computed:

\begin{pmatrix}1/64 & -1/64\\ -1/64 & 1/32\\ \end{pmatrix}

Simple basis of Cartan of centralizer (2 total):

-4h_{1}

matching e:

g_{15}-g_{-20}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2\\ \end{pmatrix}

2h_{3}+2h_{1}

matching e:

g_{14}-g_{-2}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}

Linear space basis of intersection of centralizer and ambient Cartan:

-4h_{1}

matching e:

g_{15}-g_{-20}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2\\ \end{pmatrix}

2h_{3}+2h_{1}

matching e:

g_{14}-g_{-2}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1\\ \end{pmatrix}

Elements in Cartan dual to root system: (1, 0), (-1, 0), (1, 1), (-1, -1), (1, 2), (-1, -2), (0, 1), (0, -1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form:

\begin{pmatrix}256 & -128\\ -128 & 128\\ \end{pmatrix}

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&5h_{7}+10h_{6}+10h_{5}+8h_{4}+6h_{3}+4h_{2}+2h_{1}\\ e&=&x_{1} g_{48}+x_{2} g_{40}+x_{3} g_{29}\\ f&=&x_{6} g_{-29}+x_{5} g_{-40}+x_{4} g_{-48}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(2x_{1} x_{4} +2x_{2} x_{5} +x_{3} x_{6} -5\right)h_{7}+\left(4x_{1} x_{4} +4x_{2} x_{5} +2x_{3} x_{6} -10\right)h_{6}+\left(4x_{1} x_{4} +4x_{2} x_{5} +2x_{3} x_{6} -10\right)h_{5}+\left(4x_{1} x_{4} +4x_{2} x_{5} -8\right)h_{4}+\left(4x_{1} x_{4} +2x_{2} x_{5} -6\right)h_{3}+\left(4x_{1} x_{4} -4\right)h_{2}+\left(2x_{1} x_{4} -2\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{4} -2&=&0\\4x_{1} x_{4} -4&=&0\\4x_{1} x_{4} +2x_{2} x_{5} -6&=&0\\4x_{1} x_{4} +4x_{2} x_{5} -8&=&0\\4x_{1} x_{4} +4x_{2} x_{5} +2x_{3} x_{6} -10&=&0\\4x_{1} x_{4} +4x_{2} x_{5} +2x_{3} x_{6} -10&=&0\\2x_{1} x_{4} +2x_{2} x_{5} +x_{3} x_{6} -5&=&0\\\end{array}

\begin{array}{rcl}h&=&5h_{7}+10h_{6}+10h_{5}+8h_{4}+6h_{3}+4h_{2}+2h_{1}\\e&=&x_{1} g_{48}+x_{2} g_{40}+x_{3} g_{29}\\f&=&g_{-29}+g_{-40}+g_{-48}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0\\ 4 & 0 & 0\\ 4 & 2 & 0\\ 4 & 4 & 0\\ 4 & 4 & 2\\ 4 & 4 & 2\\ 2 & 2 & 1\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}2\\ 4\\ 6\\ 8\\ 10\\ 10\\ 5\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&5h_{7}+10h_{6}+10h_{5}+8h_{4}+6h_{3}+4h_{2}+2h_{1}\\ e&=&x_{1} g_{48}+x_{2} g_{40}+x_{3} g_{29}\\ f&=&x_{6} g_{-29}+x_{5} g_{-40}+x_{4} g_{-48}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{4} -2&=&0\\4x_{1} x_{4} -4&=&0\\4x_{1} x_{4} +2x_{2} x_{5} -6&=&0\\4x_{1} x_{4} +4x_{2} x_{5} -8&=&0\\4x_{1} x_{4} +4x_{2} x_{5} +2x_{3} x_{6} -10&=&0\\4x_{1} x_{4} +4x_{2} x_{5} +2x_{3} x_{6} -10&=&0\\2x_{1} x_{4} +2x_{2} x_{5} +x_{3} x_{6} -5&=&0\\\end{array}

A^{4}_1

h-characteristic: (0, 0, 0, 1, 0, 0, 0)
Length of the weight dual to h: 8
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 3
Containing regular semisimple subalgebra number 1:

A^{2}_1+2A^{1}_1

Containing regular semisimple subalgebra number 2:

2A^{2}_1

Containing regular semisimple subalgebra number 3:

4A^{1}_1

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

10V_{2\psi}+24V_{\psi}+27V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&4h_{7}+8h_{6}+8h_{5}+8h_{4}+6h_{3}+4h_{2}+2h_{1}\\ e&=&g_{48}+g_{43}+g_{37}\\ f&=&g_{-37}+g_{-43}+g_{-48}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&4h_{7}+8h_{6}+8h_{5}+8h_{4}+6h_{3}+4h_{2}+2h_{1}\\ [h, e]&=&2g_{48}+2g_{43}+2g_{37}\\ [h, f]&=&-2g_{-37}-2g_{-43}-2g_{-48}\end{array}

Centralizer type:

C_3+2A^{4}_1

Killing form square of Cartan element dual to ambient long root: 32
Basis of the centralizer (dimension: 27):

g_{-29}

g_{-24}

g_{-19}

g_{-18}

g_{-13}

g_{-12}

g_{-7}

g_{-6}

g_{-5}

h_{1}

h_{5}

h_{6}

h_{7}

g_{2}-g_{-8}

g_{3}-g_{-3}

g_{5}

g_{6}

g_{7}

g_{8}-g_{-2}

g_{9}-g_{-14}

g_{12}

g_{13}

g_{14}-g_{-9}

g_{18}

g_{19}

g_{24}

g_{29}

Basis of centralizer intersected with cartan (dimension: 4):

-h_{1}

-h_{7}

-h_{6}

-h_{5}

Cartan of centralizer (dimension: 5):

-h_{7}

-h_{6}

-h_{5}

-h_{1}

g_{3}-g_{-3}

Cartan-generating semisimple element:

-5h_{7}+3h_{6}-2h_{5}-h_{1}

adjoint action:

\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -3/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 15/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -11/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 13 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -9 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 7/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -7/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 9 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -13 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 11/2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1/2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -15/2 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 5 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 3/2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2\\ \end{pmatrix}

Characteristic polynomial ad H:

x^{27}-793/2x^{25}+935671/16x^{23}-34165807/8x^{21}+43770315631/256x^{19}-1993647492733/512x^{17}+205109497150921/4096x^{15}-1412943297825839/4096x^{13}+4656287106059911/4096x^{11}-6283170873567549/4096x^9+588231694883925/1024x^7-4108830350625/64x^5

Factorization of characteristic polynomial of ad H: 1/4096(x )(x )(x )(x )(x )(x -13)(x -9)(x -5)(x -4)(x -2)(x +2)(x +4)(x +5)(x +9)(x +13)(2x -15)(2x -11)(2x -7)(2x -3)(2x -1)(2x -1)(2x +1)(2x +1)(2x +3)(2x +7)(2x +11)(2x +15)
Eigenvalues of ad H:

0

13

9

5

4

2

-2

-4

-5

-9

-13

15/2

11/2

7/2

3/2

1/2

-1/2

-3/2

-7/2

-11/2

-15/2

27 eigenvectors of ad H:

(0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0)

(0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0)

(0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0)

(0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0)

(0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0)

Centralizer type: C^{1}_3+2A^{4}_1
Reductive components (3 total):
Scalar product computed:

\begin{pmatrix}1/8 & -1/16 & 0\\ -1/16 & 1/16 & -1/32\\ 0 & -1/32 & 1/16\\ \end{pmatrix}

Simple basis of Cartan of centralizer (3 total):

-h_{7}-2h_{6}-2h_{5}

matching e:

g_{-29}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2\\ \end{pmatrix}

2h_{7}+4h_{6}+2h_{5}

matching e:

g_{24}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}

-2h_{7}-2h_{6}

matching e:

g_{-13}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

Linear space basis of intersection of centralizer and ambient Cartan:

-h_{7}-2h_{6}-2h_{5}

matching e:

g_{-29}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2\\ \end{pmatrix}

2h_{7}+4h_{6}+2h_{5}

matching e:

g_{24}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}-2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}

-2h_{7}-2h_{6}

matching e:

g_{-13}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

Elements in Cartan dual to root system: (1, 1, 1), (-1, -1, -1), (2, 2, 1), (-2, -2, -1), (1, 1, 0), (-1, -1, 0), (0, 0, 1), (0, 0, -1), (1, 0, 0), (-1, 0, 0), (2, 1, 1), (-2, -1, -1), (0, 1, 1), (0, -1, -1), (2, 1, 0), (-2, -1, 0), (0, 1, 0), (0, -1, 0)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form:

\begin{pmatrix}32 & -32 & 0\\ -32 & 64 & -32\\ 0 & -32 & 64\\ \end{pmatrix}

Scalar product computed:

\begin{pmatrix}1/32\\ \end{pmatrix}

Simple basis of Cartan of centralizer (1 total):

\sqrt{-1}g_{3}-2h_{1}-\sqrt{-1}g_{-3}

matching e:

g_{9}+\sqrt{-1}g_{2}-\sqrt{-1}g_{-8}-g_{-14}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & \sqrt{-1} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & \sqrt{-1} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -\sqrt{-1} & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -\sqrt{-1} & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

Linear space basis of intersection of centralizer and ambient Cartan:

\sqrt{-1}g_{3}-2h_{1}-\sqrt{-1}g_{-3}

matching e:

g_{9}+\sqrt{-1}g_{2}-\sqrt{-1}g_{-8}-g_{-14}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & \sqrt{-1} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & \sqrt{-1} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -\sqrt{-1} & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -\sqrt{-1} & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form:

\begin{pmatrix}128\\ \end{pmatrix}

Scalar product computed:

\begin{pmatrix}1/32\\ \end{pmatrix}

Simple basis of Cartan of centralizer (1 total):

-\sqrt{-1}g_{3}-2h_{1}+\sqrt{-1}g_{-3}

matching e:

g_{9}-\sqrt{-1}g_{2}+\sqrt{-1}g_{-8}-g_{-14}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & -\sqrt{-1} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & -\sqrt{-1} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{-1} & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{-1} & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

Linear space basis of intersection of centralizer and ambient Cartan:

-\sqrt{-1}g_{3}-2h_{1}+\sqrt{-1}g_{-3}

matching e:

g_{9}-\sqrt{-1}g_{2}+\sqrt{-1}g_{-8}-g_{-14}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & -\sqrt{-1} & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & -\sqrt{-1} & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{-1} & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & \sqrt{-1} & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form:

\begin{pmatrix}128\\ \end{pmatrix}

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&4h_{7}+8h_{6}+8h_{5}+8h_{4}+6h_{3}+4h_{2}+2h_{1}\\ e&=&x_{1} g_{48}+x_{2} g_{43}+x_{3} g_{37}\\ f&=&x_{6} g_{-37}+x_{5} g_{-43}+x_{4} g_{-48}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(2x_{1} x_{4} +x_{2} x_{5} +x_{3} x_{6} -4\right)h_{7}+\left(4x_{1} x_{4} +2x_{2} x_{5} +2x_{3} x_{6} -8\right)h_{6}+\left(4x_{1} x_{4} +2x_{2} x_{5} +2x_{3} x_{6} -8\right)h_{5}+\left(4x_{1} x_{4} +2x_{2} x_{5} +2x_{3} x_{6} -8\right)h_{4}+\left(4x_{1} x_{4} +2x_{2} x_{5} -6\right)h_{3}+\left(4x_{1} x_{4} -4\right)h_{2}+\left(2x_{1} x_{4} -2\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{4} -2&=&0\\4x_{1} x_{4} -4&=&0\\4x_{1} x_{4} +2x_{2} x_{5} -6&=&0\\4x_{1} x_{4} +2x_{2} x_{5} +2x_{3} x_{6} -8&=&0\\4x_{1} x_{4} +2x_{2} x_{5} +2x_{3} x_{6} -8&=&0\\4x_{1} x_{4} +2x_{2} x_{5} +2x_{3} x_{6} -8&=&0\\2x_{1} x_{4} +x_{2} x_{5} +x_{3} x_{6} -4&=&0\\\end{array}

\begin{array}{rcl}h&=&4h_{7}+8h_{6}+8h_{5}+8h_{4}+6h_{3}+4h_{2}+2h_{1}\\e&=&x_{1} g_{48}+x_{2} g_{43}+x_{3} g_{37}\\f&=&g_{-37}+g_{-43}+g_{-48}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0 & 0\\ 4 & 0 & 0\\ 4 & 2 & 0\\ 4 & 2 & 2\\ 4 & 2 & 2\\ 4 & 2 & 2\\ 2 & 1 & 1\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}2\\ 4\\ 6\\ 8\\ 8\\ 8\\ 4\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&4h_{7}+8h_{6}+8h_{5}+8h_{4}+6h_{3}+4h_{2}+2h_{1}\\ e&=&x_{1} g_{48}+x_{2} g_{43}+x_{3} g_{37}\\ f&=&x_{6} g_{-37}+x_{5} g_{-43}+x_{4} g_{-48}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{4} -2&=&0\\4x_{1} x_{4} -4&=&0\\4x_{1} x_{4} +2x_{2} x_{5} -6&=&0\\4x_{1} x_{4} +2x_{2} x_{5} +2x_{3} x_{6} -8&=&0\\4x_{1} x_{4} +2x_{2} x_{5} +2x_{3} x_{6} -8&=&0\\4x_{1} x_{4} +2x_{2} x_{5} +2x_{3} x_{6} -8&=&0\\2x_{1} x_{4} +x_{2} x_{5} +x_{3} x_{6} -4&=&0\\\end{array}

A^{3}_1

h-characteristic: (0, 0, 1, 0, 0, 0, 0)
Length of the weight dual to h: 6
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1:

A^{2}_1+A^{1}_1

Containing regular semisimple subalgebra number 2:

3A^{1}_1

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

6V_{2\psi}+24V_{\psi}+39V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&3h_{7}+6h_{6}+6h_{5}+6h_{4}+6h_{3}+4h_{2}+2h_{1}\\ e&=&g_{48}+g_{43}\\ f&=&g_{-43}+g_{-48}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&3h_{7}+6h_{6}+6h_{5}+6h_{4}+6h_{3}+4h_{2}+2h_{1}\\ [h, e]&=&2g_{48}+2g_{43}\\ [h, f]&=&-2g_{-43}-2g_{-48}\end{array}

Centralizer type:

C_4+A^{8}_1

Killing form square of Cartan element dual to ambient long root: 32
Basis of the centralizer (dimension: 39):

g_{-37}

g_{-33}

g_{-29}

g_{-28}

g_{-24}

g_{-23}

g_{-19}

g_{-18}

g_{-17}

g_{-13}

g_{-12}

g_{-11}

g_{-7}

g_{-6}

g_{-5}

g_{-4}

h_{1}

h_{4}

h_{5}

h_{6}

h_{7}

g_{2}-g_{-8}

g_{4}

g_{5}

g_{6}

g_{7}

g_{8}-g_{-2}

g_{11}

g_{12}

g_{13}

g_{17}

g_{18}

g_{19}

g_{23}

g_{24}

g_{28}

g_{29}

g_{33}

g_{37}

Basis of centralizer intersected with cartan (dimension: 5):

-h_{1}

-h_{7}

-h_{6}

-h_{5}

-h_{4}

Cartan of centralizer (dimension: 5):

-h_{7}

-h_{5}

-h_{4}

-h_{1}

-h_{6}

Cartan-generating semisimple element:

-7h_{7}+3h_{6}-6h_{5}-h_{4}-h_{1}

adjoint action:

\begin{pmatrix}1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 3 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 9 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & -9 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 11 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -6 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 17 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -13 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 7 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -7 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 13 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -17 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1/2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -5 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 6 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -4 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 8 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -11 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 9 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -9 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 4 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -5 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -3 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1\\ \end{pmatrix}

Characteristic polynomial ad H:

x^{39}-3961/4x^{37}+831081/2x^{35}-394264333/4x^{33}+14827453711x^{31}-3001513356873/2x^{29}+105614021823315x^{27}-10518711203541945/2x^{25}+186647772880296585x^{23}-18855011612675031685/4x^{21}+167786339588189504633/2x^{19}-4128440074023008475969/4x^{17}+8504546621835920723347x^{15}-44698318083894294768748x^{13}+138990557304916228362384x^{11}-225953564648336453177856x^9+149323484089920276172800x^7-25213956175372124160000x^5

Factorization of characteristic polynomial of ad H: 1/4(x )(x )(x )(x )(x )(x -17)(x -13)(x -11)(x -9)(x -9)(x -8)(x -7)(x -6)(x -5)(x -5)(x -4)(x -4)(x -3)(x -2)(x -2)(x -1)(x +1)(x +2)(x +2)(x +3)(x +4)(x +4)(x +5)(x +5)(x +6)(x +7)(x +8)(x +9)(x +9)(x +11)(x +13)(x +17)(2x -1)(2x +1)
Eigenvalues of ad H:

0

17

13

11

9

8

7

6

5

4

3

2

1

-1

-2

-3

-4

-5

-6

-7

-8

-9

-11

-13

-17

1/2

-1/2

39 eigenvectors of ad H:

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0)

(0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0)

(0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0)

(1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1)

(0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0)

(0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0)

(0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0)

(0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0)

Centralizer type: C^{1}_4+A^{8}_1
Reductive components (2 total):
Scalar product computed:

\begin{pmatrix}1/16 & 0 & 0 & -1/32\\ 0 & 1/8 & -1/16 & 0\\ 0 & -1/16 & 1/16 & -1/32\\ -1/32 & 0 & -1/32 & 1/16\\ \end{pmatrix}

Simple basis of Cartan of centralizer (4 total):

-2h_{7}-2h_{6}

matching e:

g_{-13}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 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0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -2 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & -1 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

-h_{7}-2h_{6}-2h_{5}-2h_{4}

matching e:

g_{-37}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}2 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 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2h_{4}

matching e:

g_{4}

verification:

[h,e]-2e=0

adjoint action:

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2h_{7}+4h_{6}+2h_{5}

matching e:

g_{24}

verification:

[h,e]-2e=0

adjoint action:

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Linear space basis of intersection of centralizer and ambient Cartan:

-2h_{7}-2h_{6}

matching e:

g_{-13}

verification:

[h,e]-2e=0

adjoint action:

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-h_{7}-2h_{6}-2h_{5}-2h_{4}

matching e:

g_{-37}

verification:

[h,e]-2e=0

adjoint action:

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2h_{4}

matching e:

g_{4}

verification:

[h,e]-2e=0

adjoint action:

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0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 2\\ \end{pmatrix}

2h_{7}+4h_{6}+2h_{5}

matching e:

g_{24}

verification:

[h,e]-2e=0

adjoint action:

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Elements in Cartan dual to root system: (1, 1, 1, 1), (-1, -1, -1, -1), (1, 2, 2, 2), (-1, -2, -2, -2), (1, 2, 2, 1), (-1, -2, -2, -1), (1, 2, 1, 1), (-1, -2, -1, -1), (0, 1, 1, 1), (0, -1, -1, -1), (1, 0, 1, 1), (-1, 0, -1, -1), (0, 2, 2, 1), (0, -2, -2, -1), (1, 0, 0, 1), (-1, 0, 0, -1), (0, 2, 1, 1), (0, -2, -1, -1), (0, 1, 1, 0), (0, -1, -1, 0), (1, 0, 0, 0), (-1, 0, 0, 0), (0, 0, 1, 1), (0, 0, -1, -1), (0, 2, 1, 0), (0, -2, -1, 0), (0, 0, 1, 0), (0, 0, -1, 0), (0, 0, 0, 1), (0, 0, 0, -1), (0, 1, 0, 0), (0, -1, 0, 0)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form:

\begin{pmatrix}64 & 0 & 0 & -32\\ 0 & 32 & -32 & 0\\ 0 & -32 & 64 & -32\\ -32 & 0 & -32 & 64\\ \end{pmatrix}

Scalar product computed:

\begin{pmatrix}1/64\\ \end{pmatrix}

Simple basis of Cartan of centralizer (1 total):

-4h_{1}

matching e:

g_{2}-g_{-8}

verification:

[h,e]-2e=0

adjoint action:

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& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

Linear space basis of intersection of centralizer and ambient Cartan:

-4h_{1}

matching e:

g_{2}-g_{-8}

verification:

[h,e]-2e=0

adjoint action:

\begin{pmatrix}0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 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& 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0\\ \end{pmatrix}

Elements in Cartan dual to root system: (1), (-1)
Co-symmetric Cartan Matrix of centralizer, scaled by ambient killing form:

\begin{pmatrix}256\\ \end{pmatrix}

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&3h_{7}+6h_{6}+6h_{5}+6h_{4}+6h_{3}+4h_{2}+2h_{1}\\ e&=&x_{1} g_{48}+x_{2} g_{43}\\ f&=&x_{4} g_{-43}+x_{3} g_{-48}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(2x_{1} x_{3} +x_{2} x_{4} -3\right)h_{7}+\left(4x_{1} x_{3} +2x_{2} x_{4} -6\right)h_{6}+\left(4x_{1} x_{3} +2x_{2} x_{4} -6\right)h_{5}+\left(4x_{1} x_{3} +2x_{2} x_{4} -6\right)h_{4}+\left(4x_{1} x_{3} +2x_{2} x_{4} -6\right)h_{3}+\left(4x_{1} x_{3} -4\right)h_{2}+\left(2x_{1} x_{3} -2\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{3} -2&=&0\\4x_{1} x_{3} -4&=&0\\4x_{1} x_{3} +2x_{2} x_{4} -6&=&0\\4x_{1} x_{3} +2x_{2} x_{4} -6&=&0\\4x_{1} x_{3} +2x_{2} x_{4} -6&=&0\\4x_{1} x_{3} +2x_{2} x_{4} -6&=&0\\2x_{1} x_{3} +x_{2} x_{4} -3&=&0\\\end{array}

\begin{array}{rcl}h&=&3h_{7}+6h_{6}+6h_{5}+6h_{4}+6h_{3}+4h_{2}+2h_{1}\\e&=&x_{1} g_{48}+x_{2} g_{43}\\f&=&g_{-43}+g_{-48}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0\\ 4 & 0\\ 4 & 2\\ 4 & 2\\ 4 & 2\\ 4 & 2\\ 2 & 1\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}2\\ 4\\ 6\\ 6\\ 6\\ 6\\ 3\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&3h_{7}+6h_{6}+6h_{5}+6h_{4}+6h_{3}+4h_{2}+2h_{1}\\ e&=&x_{1} g_{48}+x_{2} g_{43}\\ f&=&x_{4} g_{-43}+x_{3} g_{-48}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{3} -2&=&0\\4x_{1} x_{3} -4&=&0\\4x_{1} x_{3} +2x_{2} x_{4} -6&=&0\\4x_{1} x_{3} +2x_{2} x_{4} -6&=&0\\4x_{1} x_{3} +2x_{2} x_{4} -6&=&0\\4x_{1} x_{3} +2x_{2} x_{4} -6&=&0\\2x_{1} x_{3} +x_{2} x_{4} -3&=&0\\\end{array}

A^{2}_1

h-characteristic: (0, 1, 0, 0, 0, 0, 0)
Length of the weight dual to h: 4
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Number of containing regular semisimple subalgebras: 2
Containing regular semisimple subalgebra number 1:

2A^{1}_1

Containing regular semisimple subalgebra number 2:

A^{2}_1

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

3V_{2\psi}+20V_{\psi}+56V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&2h_{7}+4h_{6}+4h_{5}+4h_{4}+4h_{3}+4h_{2}+2h_{1}\\ e&=&g_{49}+g_{47}\\ f&=&g_{-47}+g_{-49}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&2h_{7}+4h_{6}+4h_{5}+4h_{4}+4h_{3}+4h_{2}+2h_{1}\\ [h, e]&=&2g_{49}+2g_{47}\\ [h, f]&=&-2g_{-47}-2g_{-49}\end{array}

Centralizer type:

C_5

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&2h_{7}+4h_{6}+4h_{5}+4h_{4}+4h_{3}+4h_{2}+2h_{1}\\ e&=&x_{1} g_{49}+x_{2} g_{47}\\ f&=&x_{4} g_{-47}+x_{3} g_{-49}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(x_{1} x_{3} +x_{2} x_{4} -2\right)h_{7}+\left(2x_{1} x_{3} +2x_{2} x_{4} -4\right)h_{6}+\left(2x_{1} x_{3} +2x_{2} x_{4} -4\right)h_{5}+\left(2x_{1} x_{3} +2x_{2} x_{4} -4\right)h_{4}+\left(2x_{1} x_{3} +2x_{2} x_{4} -4\right)h_{3}+\left(2x_{1} x_{3} +2x_{2} x_{4} -4\right)h_{2}+\left(2x_{1} x_{3} -2\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{3} -2&=&0\\2x_{1} x_{3} +2x_{2} x_{4} -4&=&0\\2x_{1} x_{3} +2x_{2} x_{4} -4&=&0\\2x_{1} x_{3} +2x_{2} x_{4} -4&=&0\\2x_{1} x_{3} +2x_{2} x_{4} -4&=&0\\2x_{1} x_{3} +2x_{2} x_{4} -4&=&0\\x_{1} x_{3} +x_{2} x_{4} -2&=&0\\\end{array}

\begin{array}{rcl}h&=&2h_{7}+4h_{6}+4h_{5}+4h_{4}+4h_{3}+4h_{2}+2h_{1}\\e&=&x_{1} g_{49}+x_{2} g_{47}\\f&=&g_{-47}+g_{-49}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2 & 0\\ 2 & 2\\ 2 & 2\\ 2 & 2\\ 2 & 2\\ 2 & 2\\ 1 & 1\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}2\\ 4\\ 4\\ 4\\ 4\\ 4\\ 2\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&2h_{7}+4h_{6}+4h_{5}+4h_{4}+4h_{3}+4h_{2}+2h_{1}\\ e&=&x_{1} g_{49}+x_{2} g_{47}\\ f&=&x_{4} g_{-47}+x_{3} g_{-49}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{3} -2&=&0\\2x_{1} x_{3} +2x_{2} x_{4} -4&=&0\\2x_{1} x_{3} +2x_{2} x_{4} -4&=&0\\2x_{1} x_{3} +2x_{2} x_{4} -4&=&0\\2x_{1} x_{3} +2x_{2} x_{4} -4&=&0\\2x_{1} x_{3} +2x_{2} x_{4} -4&=&0\\x_{1} x_{3} +x_{2} x_{4} -2&=&0\\\end{array}

A_1

h-characteristic: (1, 0, 0, 0, 0, 0, 0)
Length of the weight dual to h: 2
Simple basis ambient algebra w.r.t defining h: 7 vectors: (1, 0, 0, 0, 0, 0, 0), (0, 1, 0, 0, 0, 0, 0), (0, 0, 1, 0, 0, 0, 0), (0, 0, 0, 1, 0, 0, 0), (0, 0, 0, 0, 1, 0, 0), (0, 0, 0, 0, 0, 1, 0), (0, 0, 0, 0, 0, 0, 1)
Containing regular semisimple subalgebra number 1:

A^{1}_1

sl{}\left(2\right)

-module decomposition of the ambient Lie algebra:

V_{2\psi}+12V_{\psi}+78V_{0}

Below is one possible realization of the sl(2) subalgebra.

\begin{array}{rcl}h&=&h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}+2h_{2}+2h_{1}\\ e&=&g_{49}\\ f&=&g_{-49}\end{array}

Lie brackets of the above elements.

\begin{array}{rcl}[e, f]&=&h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}+2h_{2}+2h_{1}\\ [h, e]&=&2g_{49}\\ [h, f]&=&-2g_{-49}\end{array}

Centralizer type:

C_6

Unfold the hidden panel for more information.

Unknown elements.

\begin{array}{rcl}h&=&h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}+2h_{2}+2h_{1}\\ e&=&x_{1} g_{49}\\ f&=&x_{2} g_{-49}\end{array}

Participating positive roots: 0 vectors. .
Lie brackets of the unknowns.

[e,f] - h = \left(x_{1} x_{2} -1\right)h_{7}+\left(2x_{1} x_{2} -2\right)h_{6}+\left(2x_{1} x_{2} -2\right)h_{5}+\left(2x_{1} x_{2} -2\right)h_{4}+\left(2x_{1} x_{2} -2\right)h_{3}+\left(2x_{1} x_{2} -2\right)h_{2}+\left(2x_{1} x_{2} -2\right)h_{1}

The polynomial system that corresponds to finding the h, e, f triple:

\begin{array}{rcl}2x_{1} x_{2} -2&=&0\\2x_{1} x_{2} -2&=&0\\2x_{1} x_{2} -2&=&0\\2x_{1} x_{2} -2&=&0\\2x_{1} x_{2} -2&=&0\\2x_{1} x_{2} -2&=&0\\x_{1} x_{2} -1&=&0\\\end{array}

\begin{array}{rcl}h&=&h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}+2h_{2}+2h_{1}\\e&=&x_{1} g_{49}\\f&=&g_{-49}\end{array}

Matrix form of the system we are trying to solve:

\begin{pmatrix}2\\ 2\\ 2\\ 2\\ 2\\ 2\\ 1\\ \end{pmatrix}[col. vect.]=\begin{pmatrix}2\\ 2\\ 2\\ 2\\ 2\\ 2\\ 1\\ \end{pmatrix}

The unknown Kostant-Sekiguchi elements.

\begin{array}{rcl}h&=&h_{7}+2h_{6}+2h_{5}+2h_{4}+2h_{3}+2h_{2}+2h_{1}\\ e&=&x_{1} g_{49}\\ f&=&x_{2} g_{-49}\end{array}

e-f=0

\theta(e-f)=0

The polynomial system we need to solve.

\begin{array}{rcl}2x_{1} x_{2} -2&=&0\\2x_{1} x_{2} -2&=&0\\2x_{1} x_{2} -2&=&0\\2x_{1} x_{2} -2&=&0\\2x_{1} x_{2} -2&=&0\\2x_{1} x_{2} -2&=&0\\x_{1} x_{2} -1&=&0\\\end{array}

sl(2)sl{}\left(2\right)-subalgebras of sp(14), type C7C_7

$sl{}\left(2\right)$ -subalgebras of sp(14), type $C_7$